1 2Dipartimento di Fisica Nucleare e Teorica, Universita ...2Dipartimento di Fisica Nucleare e...

arX

iv:0

807.

0323

v2 [

hep-

ph]

11

Jul 2

008

JLAB-THY-08-841

Transverse-momentum distributions in a diquark spectator model

Alessandro Bacchetta,1, ∗ Francesco Conti,2, 3, † and Marco Radici3, ‡

1Theory Center, Jefferson Lab, 12000 Jefferson Av., Newport News, VA 23606, USA2Dipartimento di Fisica Nucleare e Teorica, Universita di Pavia, I-27100 Pavia, Italy

3Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy

All the leading-twist parton distribution functions are calculated in a spectator model of thenucleon, using scalar and axial-vector diquarks. Single gluon rescattering is used to generate T-odddistribution functions. Different choices for the diquark polarization states are considered, as wellas a few options for the form factor at the nucleon-quark-diquark vertex. The results are listedin analytic form and interpreted in terms of light-cone wave functions. The model parametersare fixed by reproducing the phenomenological parametrization of unpolarized and helicity partondistributions at the lowest available scale. Predictions for the other parton densities are given and,whenever possible, compared with available phenomenological parametrizations.

PACS numbers: 12.39.-x, 13.60.-r, 13.88.+e

I. INTRODUCTION

Partonic transverse-momentum distributions (TMDs) — also called unintegrated PDFs — describe the probabilityto find in a hadron a parton with longitudinal momentum fraction x and transverse momentum pT with respect tothe direction of the parent hadron momentum [1]. They give a three-dimensional view of the parton distribution inmomentum space, complementary to what can be obtained through generalized parton distributions [2, 3, 4, 5, 6].In the last years a lot of theoretical and experimental activity related to TMDs has taken place. Crucial steps were

made in the understanding of factorization theorems involving TMDs (kT factorization)[7, 8]. Some of the propertiesof TMDs have been investigated from the theoretical standpoint. For instance, positivity bounds were presented inRef. [9]. Relations among these functions in the large-Nc limit of QCD were put forward in Ref. [10]. Their behaviorat large x was studied in Ref. [11], and at high transverse momentum in Ref. [12]. Last but not least, it was alsodemonstrated [13] that TMDs that are odd under naıve time-reversal transformations (for brevity, T-odd) can benonzero and must be included in the complete list of leading-twist TMDs (see, e.g., Ref. [14, 15]). Their universalityproperties are different from the standard PDFs [16].In the meanwhile, several azimuthal asymmetries were measured in semi-inclusive deep inelastic scattering (SIDIS)

and elsewhere (see Ref. [17] and references therein), and more experimental measurements are planned. However, notmuch phenomenological information concerning TMDs is available as yet (see, e.g., Ref. [18] and references therein).The analysis of azimuthal spin asymmetries both in hadron-hadron collisions and in SIDIS led to the extraction ofthe Sivers function [19], denoted as f⊥

1T , a T-odd TMD that describes how the parton distribution is distorted bythe transverse polarization of the parent hadron (see Ref. [20] for a comparison of various parametrizations). Arecent attempt to extract the T-odd Boer-Mulders function, h⊥1 [21], a T-odd TMD describing the distribution oftransversely polarized partons in an unpolarized hadron, was presented in Ref. [22]. All of the above studies assumea flavor-independent Gaussian distribution of the transverse momentum, although there is no compelling reason forthis choice.In this context, building a relatively simple model to compute TMDs and to allow for numerical estimates is of

great importance. From the theoretical side, this can help understanding some of the essential features of TMDs, forinstance their relation to the orbital angular momentum of partons (see, e.g., Refs. [13, 23, 24, 25, 26, 27, 28, 29]).From the experimental side, a model could be useful to estimate the size of observables in different processes andkinematical regimes [30, 31, 32, 33, 34, 35, 36] and to set up Monte Carlo simulations [37, 38, 39, 40, 41].Although many model calculations of integrated PDFs are available, there are not so many for TMDs. In Ref. [42]

all the leading-twist T-even functions were calculated in a spectator model with scalar and axial-vector diquarks.Recently, an analogous calculation has been performed in a light-cone quark model [43]. T-odd functions werecalculated in the spectator model with scalar diquarks [13, 30, 44, 45], with scalar and vector diquarks [36, 46], in the

∗Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]

http://arxiv.org/abs/0807.0323v2

mailto:[email protected]



2

MIT bag model [47, 48], in a constituent quark model [49] and in the spectator model for the pion [50]. A completecalculation of all the leading-twist TMDs in a spectator model with scalar diquarks was presented in Ref. [27].In this work, we choose a more phenomenological approach. We consider also axial-vector diquarks (in the following

often called simply vector diquarks), necessary for a realistic flavor analysis, and we further distinguish betweenisoscalar (ud-like) and isovector (uu-like) spectators. We generate the relative phase necessary to produce T-oddstructures by approximating the gauge link operator with a one gluon-exchange interaction. We consider severalchoices of form factors at the nucleon-quark-diquark vertex and several choices for the polarization states of thediquark. All results are presented in analytic form and interpreted also in terms of overlaps of light-cone wavefunctions, leading to a detailed analysis of the quantum numbers of the quark-diquark system. The free parameters ofthe model are fixed by reproducing the phenomenological parametrization of unpolarized and longitudinally polarizedparton distributions at the lowest available scale.The paper is organized as follows. In Sec. II, the analytic form for all the leading-twist TMDs is discussed for

the dipolar nucleon-diquark-quark form factor and for the light-cone choice of the diquark propagator, postponingthe results for the other explored combinations to the Appendices A (T-even TMDs) and B (T-odd TMDs). InSec. III, numerical results are shown and compared with phenomenological parametrizations, whenever available inthe literature. In Sec. IV, some conclusions are drawn.

II. ANALYTICAL RESULTS FOR TRANSVERSE-MOMENTUM-DEPENDENT PARTON DENSITIES

In this section we present the fundamentals of the model and we give in analytical form the results for the light-conewave functions (LCWFs) and the TMDs obtained in the model.

A. General framework

In the following we will make use of light-cone coordinates. We introduce the light-like vectors n± satisfyingn2± = 0, n+ · n− = 1, and we describe a generic 4-vector a as

a = [a−, a+,aT ] (1)

where a± = a ·n∓. We will make use of the transverse tensor ǫijT = ǫµνijn+µn−ν , whose only nonzero components areǫ12T

= −ǫ21T

= 1. We choose a frame where the hadron momentum P has no transverse components, i.e.,

P =

[

M2

2P+, P+,0

]

. (2)

The quark momentum can be written as

p =

[

p2 + p2T

2xP+, xP+,pT

]

. (3)

In a hadronic state |P, S〉 with momentum P and spin S, the density of quarks can be defined starting from thequark-quark correlator (see, e.g., Ref. [15])

Φ(x,pT ;S) =

∫

dξ−dξT

(2π)3eip·ξ 〈P, S|ψ(0)U[0,ξ] ψ(ξ)|P, S〉

∣

∣

∣

ξ+=0, (4)

where

U[0,ξ] = P e−igR

ξ

0dw·A(w) (5)

is the so-called gauge link operator, or Wilson line, connecting the two different space-time points 0, ξ, by all possibleordered paths followed by the gluon field A, which couples to the quark field ψ through the coupling g. The gaugelink ensures that the matrix element of Eq. (4) is color-gauge invariant and arises from the interaction of the outgoingquark field with the spectators inside the hadron. The leading contributions of the path [0, ξ] in space-time areselected by the hard process in which the parton distributions appear, thus breaking standard universality of theparton densities. For instance, in SIDIS the gauge link path in light-cone coordinates runs along

[0, ξ] ≡ (0, 0,0T ) → (0,∞,0T ) → (0,∞,∞T ) → (0,∞, ξT ) → (0, ξ−, ξT ) , (6)

3

while in the Drell–Yan case it runs in the opposite direction through −∞. This fact leads to a sign difference in T-oddparton densities, as mentioned for the first time in Ref. [16].Similarly to Ref. [42], we evaluate the correlator of Eq. (4) in the spectator approximation, i.e. we insert a com-

pleteness relation and at tree-level we truncate the sum over final states to a single on-shell spectator state with massMX , thus getting the analytic form

Φ(x,pT , S) ∼1

(2π)31

2(1− x)P+M(0)

(S)M(0)(S)∣

∣

∣

p2=τ(x,pT), (7)

where p is the momentum of the active quark, m its mass, and the on-shell condition (P −p)2 =M2X for the spectator

implies for the quark the off-shell condition

p2 ≡ τ(x,pT ) = −p2T+ L2

X(m2)

1− x+m2 , L2

X(m2) = xM2X + (1− x)m2 − x(1− x)M2 , (8)

with M the hadron mass.

P

p

p − PY

FIG. 1: Tree-level cut diagram for the calculation of T-even leading-twist parton densities. The dashed line indicates bothscalar and axial-vector diquarks.

We assume the spectator to be point-like, with the quantum numbers of a diquark. Hence, the proton can couple toa quark and to a spectator diquark with spin 0 (scalar X = s) or spin 1 (axial-vector X = a), as well as with isospin0 (isoscalar ud-like system) or isospin 1 (isovector uu-like system). Therefore, the tree-level “scattering amplitude”M(0) is given by (see Fig. 1)

M(0)(S) = 〈P − p|ψ(0)|P, S〉 =

i

p/−mYs U(P, S) scalar diquark,

i

p/−mε∗µ(P − p, λa)Yµ

a U(P, S) axial-vector diquark,(9)

and is actually a Dirac spinor because of the understood spinorial indices of the quark field ψ. The εµ(P − p, λa) isthe 4-vector polarization of the spin-1 vector diquark with momentum P − p and helicity states λa. When summingover all polarizations states, several choices have been used for dµν =

∑

λaε∗µ(λa)

εν(λa):

dµν(P − p) =

−gµν + (P − p)µnν− + (P − p)νnµ

−

(P − p) · n−− M2

a

[(P − p) · n−]2nµ− n

ν− (see Ref. [51]),

−gµν + (P − p)µ (P − p)ν

M2a

(see Ref. [36]),

−gµν +Pµ P ν

M2a

(see Ref. [42]),

−gµν (see Ref. [46]).

(10)

The different forms for the diquark propagator correspond to different physical theories and lead to different resultsfor the parton distribution functions. We have analyzed all of them except for the third one, which was extensivelystudied already in Ref. [42]. However, we think that the first one is preferable to the others. The motivation isthat in the spectator model we have to take into account that the diquarks have an electric charge and can coupleto the virtual photon in DIS. In other words, in this model the quarks are not the only charged partons in theproton: the diquarks are also charged partons and they have spin different from 1

2 . The scalar diquark couples only

4

to longitudinally polarized photons and gives contribution to the structure function FL. This leads to a violationof the Callan–Gross relation, but leaves unchanged the (leading-order) interpretation of the structure function FT

as a charge-weighted sum of quark distribution functions. This seems the best way to reduce the phenomenologicalimpact of the problem represented by the presence of the diquarks. For the vector diquark, we checked that the samesituation occurs when only the light-cone transverse polarization states of the diquark are propagated, i.e., when thefirst choice of Eq. (10) for the polarization sum is used. In the other cases, the diquark would give a contributionalso to the structure function FT . On top of this, we remark that the last choice of Eq. (10) for the polarization sumintroduces unphysical polarization states of the vector diquark (see discussion in next section). In conclusion, in thefollowing we shall consider only light-cone transverse polarizations of the diquark, make only a few comments on theother choices, and leave the complete list of results in the Appendices.

P

p

p − PY

FIG. 2: Tree-level cut diagram for the calculation of T-even leading-twist parton densities for an active scalar or vector diquark(dashed line), with a spectator quark (solid line).

Equation (9) can be further elaborated by choosing the nucleon-quark-diquark vertex Y. We choose the scalar andvector vertices to be

Ys = igs(p2) 1l , Yµ

a = iga(p

2)√2

γµ γ5 , (11)

where gX(p2) is a suitable form factor. Other choices are possible (see, e.g., Refs [36, 42]), but we limit ourselves tothese ones, which are the simplest. For the form factor, we explored three possible choices:

gX(p2) =

gp.l.X point-like,

gdipX

p2 −m2

|p2 − Λ2X |2

dipolar,

gexpX e (p2−m2)/Λ2X exponential,

(12)

where gX and ΛX are appropriate coupling constants and cutoffs, respectively, to be considered as free parametersof the model together with the mass of the diquark MX . All these parameters can in principle be different for eachtype of diquark. Only the point-like coupling can be derived from a specific Lagrangian with protons, quarks anddiquarks as fundamental degrees of freedom, and meant to effectively describe QCD in the nonperturbative regime.Since our interest here is mainly phenomenological, we prefer to introduce form factors. They smoothly suppress theinfluence of high pT — where our theory cannot be trusted — and eliminate the logarithmic divergences arising afterpT integration when using a point-like coupling. For later use, we note that the dipolar form factor can be usefullyrewritten, using Eq. (8), as

gX(p2) = gdipX

p2 −m2

|p2 − Λ2X |2

= gdipX

(p2 −m2) (1 − x)2

(p2T+ L2

X(Λ2X))

2 . (13)

In summary, we have analyzed in total nine combinations of nucleon-quark-diquark form factors and forms for thediquark propagator. As mentioned above, we will discuss analytical and numerical results involving the dipolar formfactor and the first choice of Eq. (10) (transverse diquark polarizations only), listing the formulae for the other cases

in the Appendices A and B. To keep the notation lighter, we will denote the coupling gdipX simply as gX from now on.

5

B. Light-cone wave functions

A convenient way to compute parton distribution functions is by making use of light-cone wave functions (LCWFs),as done for instance in Ref. [51]. For the scalar diquark, LCWFs can be defined as

ψλN

λq(x,pT ) =

√

p+

(P − p)+u(p, λq)

p2 −m2Ys U(P, λN ) , (14)

where the indices λN , λq , refer to the helicity of the nucleon and of the quark, respectively, and are constrained byangular momentum conservation to the “spin sum rule” λN = λq + Lz, where Lz is the projection of the relativeorbital angular momentum between the quark and the diquark. We use the conventions of Ref. [52] (see also Ref. [27]).In standard representation, the spinors can be written as

u(p,+) =1

√

23/2 p+

√2 p+ +mpx + ipy√2 p+ −mpx + ipy

, u(p,−) =1

√

23/2 p+

−px + ipy√2 p+ +mpx − ipy

−√2 p+ +m

, (15)

and similarly for the nucleon spinors (changing p,m, to P,M , respectively). We obtain

ψ++(x,pT ) = (m+ xM)φ/x (Lz = 0), (16)

ψ+−(x,pT ) = −(px + ipy)φ/x (Lz = +1), (17)

ψ−+(x,pT ) = −

[

ψ+−(x,pT )

]∗(Lz = −1), (18)

ψ−−(x,pT ) = ψ+

+(x,pT ) (Lz = 0), (19)

φ(x,p2T) = − gs√

1− x

x(1− x)

p2T+ L2

s(m2), (20)

which correspond to Eqs. (44,46) of Ref. [51].For the vector diquark, LCWFs can be defined as

ψλN

λqλa(x,pT ) =

√

p+

(P − p)+u(p, λq)

p2 −m2ε∗µ(P − p, λa)Yµ

a U(P, λN ) , (21)

where the index λa refers to the helicity of the vector diquark and is constrained by λN = λq+λa+Lz. The light-conetransverse polarization vectors are given by [51]

ε(P − p,+) =

[

− (P − p)x + i(P − p)y√2 (P − p)+

, 0,− 1√2,− i√

2

]

=

[

px + ipy√2 (1− x)P+

, 0,− 1√2,− i√

2

]

, (22)

ε(P − p,−) =

[

− px − ipy√2 (1− x)P+

, 0,1√2,− i√

2

]

. (23)

They satisfy the usual properties1 ε(±) · ε∗(±) = −1, ε(±) · ε∗(∓) = 0, and (P − p) · ε(±) = 0. They are consistentwith the polarization sum being expressed by the first option in Eq. (10). The LCWFs become

ψ+++(x,pT ) =

px − ipy1− x

φ/x (Lz = −1), (24)

ψ++−(x,pT ) = −x px + ipy

1− xφ/x (Lz = +1), (25)

ψ+−+(x,pT ) = (m+ xM)φ/x (Lz = 0), (26)

ψ+−−(x,pT ) = 0 (Lz = +2), (27)

ψ−++(x,pT ) = 0 (Lz = −2), (28)

1 Note that (P − p) · ε(±) 6= 0, since ε(±) do not describe transverse polarization with respect to the diquark momentum.

6

ψ−+−(x,pT ) = −ψ+

−+(x,pT ) (Lz = 0), (29)

ψ−−+(x,pT ) =

[

ψ++−(x,pT )

]∗(Lz = −1), (30)

ψ−−−(x,pT ) =

[

ψ+++(x,pT )

]∗(Lz = +1), (31)

φ(x,p2T) = − ga√

1− x

x(1− x)

p2T+ L2

a(m2), (32)

and are analogous to Eqs. (21,24) in Ref. [51], the differences being due to the fact that here the diquark is anaxial-vector particle rather than a vector one. Note that in our model we can only have wavefunctions with at mostone unit of orbital angular momentum (p wave). The LCWFs with two units of orbital angular momentum (d-wave),ψ+−− and ψ−

++, vanish.If we add to ε(P − p,±) also the third longitudinal polarization vector

ε(P − p, 0) =1

Ma

[

p2T−M2

a

2 (1− x)P+, (1− x)P+,−px,−py

]

, (33)

satisfying2 ε(0) · ε∗(0) = −1, ε(0) · ε∗(±) = 0, and (P − p) · ε(0) = 0, the corresponding additional LCWFs are

ψ++0(x,pT ) =

p2T− xM2

a −mM (1− x)2√2 (1− x)Ma

φ/x (Lz = 0), (34)

ψ+−0(x,pT ) =

(m+M) (px + ipy)√2Ma

φ/x (Lz = +1), (35)

ψ−+0(x,pT ) =

[

ψ+−0(x,pT )

]∗φ/x (Lz = −1), (36)

ψ−−0(x,pT ) = −ψ+

+0(x,pT )φ/x (Lz = 0). (37)

From the above combinations we deduce, for example, that the proton with positive helicity + 12 can be in a state

with probability density proportional to |ψ+−|2, where the quark has opposite helicity and Lz = +1 orbital angular

momentum with respect to a scalar diquark. This configuration is relativistically enhanced with respect to |ψ++ |2 with

Lz = 0, where proton and quark helicities are aligned; thus, it suggests a possible explanation of the proton “spinpuzzle” in terms of the relativistic aspects of the motion of quarks inside hadrons [51].For the purpose of this work, it is also important to note that a nonvanishing relative orbital angular momentum

between the quark and the diquark implies that the partons do not necessarily occupy the lowest-energy available

orbital (with quantum numbers JP = 12

+and Lz = 0). Hence, in this version of the spectator diquark model

the nucleon wave function does not show a SU(4)=SU(2)⊗SU(2) spin-isospin symmetry, contrary to what is usuallyassumed [42].Finally, we mention that the completeness relation for the last choice of the polarization sum in Eq. (10) should be

written∑

λa=±,0

ε∗µ(P − p, λa) εν(P − p, λa)− ε∗µ(P − p, t) εν(P − p, t) = −gµν , (38)

where the unphysical time-like polarization state εµ(P − p, t) = (P − p)µ/Ma appears. The associated LCWFs read

ψ++t(x,pT ) =

p2T+ xM2

a −mM (1 − x)2√2 (1 + x)Ma

φ/x (Lz = 0), (39)

ψ+−t(x,pT ) =

(m+M) (px + ipy)√2Ma

φ/x (Lz = +1), (40)

ψ−+t(x,pT ) =

[

ψ+−t(x,pT )

]∗φ/x (Lz = −1), (41)

ψ−−t(x,pT ) = −ψ+

+t(x,pT )φ/x (Lz = 0). (42)

2 Note that ε(0) is not parallel to (P − p) because it describes longitudinal polarization states in the light-cone.

7

C. T-even functions

The simplest example of T-even parton density is the unpolarized quark distribution f1(x,pT ), defined as

f1(x,pT ) =1

4Tr

[

(Φ(x,pT , S) + Φ(x,pT ,−S)) γ+]

+ h.c.

=1

4

1

(2π)31

2(1− x)P+Tr

[(

M(0)(S)M(0)(S) +M(0)

(−S)M(0)(−S))

γ+]

+ h.c. .(43)

By inserting in M(0) of Eq. (9) the rules (11) for the nucleon-quark-diquark vertex, the dipolar form factor of Eq. (13),and the first choice in Eq. (10) for the sum of the polarization states of the diquark (transverse polarizations only),we get

fq(s)1 (x,pT ) =

g2s(2π)3

[(m+ xM)2 + p2T] (1− x)3

2 [p2T+ L2

s(Λ2s)]

4(44)

fq(a)1 (x,pT ) =

g2a(2π)3

[p2T(1 + x2) + (m+ xM)2 (1− x)2] (1 − x)

2 [p2T+ L2

a(Λ2a)]

4. (45)

The same result can be recovered through the alternative definition

fq(s)1 (x,p2

T) =

1

16π3

1

2

∑

λN=±

∑

λq=±

|ψλN

λq|2 =

1

16π3

(

|ψ++ |2 + |ψ+

−|2)

(46)

fq(a)1 (x,p2

T) =

1

16π3

1

2

∑

λN=±

∑

λq=±

∑

λa=±

|ψλN

λqλa|2 =

1

16π3

(

|ψ+++|2 + |ψ+

+−|2 + |ψ+−+|2 + |ψ+

−−|2)

, (47)

and replacing the results for the LCWFs using Eqs. (20) and (32) for the scalar and vector diquark, respectively.If we use, instead, the second option of Eq. (10) for the sum over polarizations of the vector diquark (transverse

and longitudinal polarizations), we obtain

fq(a)1 (x,pT ) +

1

16π3

(

|ψ++0|2 + |ψ+

−0|2)

. (48)

The complete expression is given in Eq. (A22) and corresponds to Eq. (10) of Ref. [36] with Rg = 0.Finally, the results with the last choice of Eq. (10) (transverse, longitudinal, and time-like polarizations) can be

written as

fq(a)1 (x,pT ) +

1

16π3

(

|ψ++0|2 + |ψ+

−0|2)

− 1

16π3

(

|ψ++t|2 + |ψ+

−t|2)

. (49)

Note that the contribution of the diquark time-like polarization states enters with an overall negative sign. Thecomplete expression is given in Eq. (A26) and corresponds to Eq. (8) of Ref. [46].Turning back to our preferred choice, i.e. the first option of Eq. (10) (light-cone transverse polarizations only), we

now compute all other T-even, leading-twist TMDs. Their definition in terms of traces of the quark-quark correlatorcan be derived from, e.g., Eqs. (3.19) and ff. in Ref. [15]. To write them in terms of LCWFs, we need to introduce

the polarization state in a generic direction ST = (cosφS , sinφS) in the transverse plane

U(P, ↑) = 1√2

(

U(P,+) + eiφSU(P,−))

, (50)

U(P, ↓) = 1√2

(

U(P,+) + ei(φS+π)U(P,−))

. (51)

For φS = 0, π/2, we recover the (positive) polarizations along the x and y axis, respectively [53]. For the quark, we

will use similar decompositions and use the notation SqT and φSq, i.e.,

u(p, ↑) = 1√2

(

u(p,+) + e−iφSq u(p,−))

, (52)

u(p, ↓) = 1√2

(

u(p,+) + e−i(φSq+π)u(p,−))

. (53)

8

With this conventions and keeping in mind that λX is absent for the scalar diquark and λX = ± for the vectordiquark, we can write the TMDs in the following way

g1L(x,pT ) =1

16π3

∑

λX

(

|ψ++λX

|2 − |ψ+−λX

|2)

, (54)

pT · ST

Mg1T (x,pT ) =

1

16π3

∑

λX

(

|ψ↑+λX

|2 − |ψ↑−λX

|2)

(55)

pT · SqT

Mh⊥1L(x,pT ) =

1

16π3

∑

λX

(

|ψ+↑λX

|2 − |ψ+↓λX

|2)

, (56)

ST · SqT h1T (x,pT ) +pT · ST

M

pT · SqT

Mh⊥1T (x,pT ) =

1

16π3

∑

λX

(

|ψ↑↑λX

|2 − |ψ↑↓λX

|2)

. (57)

The above results automatically fulfill positivity bounds [9].The explicit expressions are

gq(s)1L (x,p2

T) =

g2s(2π)3

[(m+ xM)2 − p2T] (1− x)3

2 [p2T+ L2

s(Λ2s)]

4, (58)

gq(a)1L (x,p2

T) =

g2a(2π)3

[p2T(1 + x2)− (m+ xM)2 (1− x)2] (1− x)

2 [p2T+ L2

a(Λ2a)]

4, (59)

gq(s)1T (x,p2

T) =

g2s(2π)3

M (m+ xM) (1 − x)3

[p2T+ L2

s(Λ2s)]

4, (60)

gq(a)1T (x,p2

T) =

g2a(2π)3

xM (m+ xM) (1− x)2

[p2T+ L2

a(Λ2a)]

4, (61)

h⊥ q(s)1L (x,p2

T) = − g2s

(2π)3M (m+ xM) (1− x)3

[p2T+ L2

s(Λ2s)]

4, (62)

h⊥ q(a)1L (x,p2

T) =

g2a(2π)3

M (m+ xM) (1 − x)2

[p2T+ L2

a(Λ2a)]

4, (63)

hq(s)1T (x,p2

T) =

g2s(2π)3

[p2T+ (m+ xM)2] (1− x)3

2 [p2T+ L2

s(Λ2s)]

4, (64)

hq(a)1T (x,p2

T) = − g2a

(2π)3p2

Tx(1 − x)

[p2T+ L2

a(Λ2a)]

4, (65)

h⊥ q(s)1T (x,p2

T) = − g2s

(2π)3M2 (1− x)3

[p2T+ L2

s(Λ2s)]

4, (66)

h⊥ q(a)1T (x,p2

T) = 0 . (67)

From the last two formulae we deduce also the expressions for the transversity distribution:

hq(s)1 (x,p2

T) = h

q(s)1T (x,p2

T) +

p2T

2M2h⊥ q(s)1T (x,p2

T) =

g2s(2π)3

(m+ xM)2 (1 − x)3

2 [p2T+ L2

s(Λ2s)]

4(68)

hq(a)1 (x,p2

T) = − g2a

(2π)3p2

Tx(1 − x)

[p2T+ L2

a(Λ2a)]

4. (69)

Note that the functions g1T and h⊥1L arise from the interference of LCWFs with |Lz| = 1 and Lz = 0. The function h⊥1Trequires the interference of two LCWFs that differ by two units of Lz. This condition is necessary but not sufficientto have h⊥1T 6= 0. In fact, the vector diquark spectator gives h⊥1T = 0 even if LCWFs with Lz = ±1 are present.

9

Some interesting relations can be evinced from the above expressions. For example, the transversity with scalardiquark saturates the Soffer bound, while for axial-vector diquarks the relation is more involved:

hq(s)1 (x,p2

T) =

1

2

(

fq(s)1 (x,p2

T) + g

q(s)1 (x,p2

T))

, (70)

hq(a)1 (x,p2

T) = − x

1 + x21

2

(

fq(a)1 (x,p2

T) + g

q(a)1 (x,p2

T))

. (71)

When restricting to the results with scalar diquark, the g1T distribution is connected to two other partners by therelations

gq(s)1T (x,p2

T) = −h⊥ q(s)

1L (x,p2T) , g

q(s)1T (x,p2

T) =

2M

m+ xMhq(s)1 (x,p2

T) , (72)

while for axial-vector diquarks we have

gq(a)1T (x,p2

T) = xh

⊥ q(a)1L (x,p2

T) . (73)

This relation is however different when considering spectator diquarks with more degrees of freedom (see App. A).Our results seem to indicate that no general relation exists between g1T and h⊥1L, contrary to what is proposed inRef. [43]. The reason is connected to the difference between LCWFs with Lz = 1 and Lz = −1, as in Eqs. (24) and(25). We also observe that in the vector-diquark case g1L − h1 and h⊥1T are not simply related through the relationsuggested in Ref. [54]. We are led to conclude that such a relation is not general.The pT -integrated results are

fq(s)1 (x) =

g2s(2π)2

[2 (m+ xM)2 + L2s(Λ

2s)] (1− x)3

24L6s(Λ

2s)

(74)

fq(a)1 (x) =

g2a(2π)2

[2 (m+ xM)2 (1 − x)2 + (1 + x2)L2a(Λ

2a)] (1− x)

24L6a(Λ

2a)

(75)

gq(s)1 (x) =

g2s(2π)2

[2 (m+ xM)2 − L2s(Λ

2s)] (1− x)3

24L6s(Λ

2s)

(76)

gq(a)1 (x) = − g2a

(2π)2[2 (m+ xM)2 (1− x)2 − (1 + x2)L2

a(Λ2a)] (1 − x)

24L6a(Λ

2a)

(77)

hq(s)1 (x) =

g2s(2π)2

(m+ xM)2 (1− x)3

12L6s(Λ

2s)

(78)

hq(a)1 (x) = − g2a

(2π)2x(1 − x)

12L4a(Λ

2a). (79)

D. T-odd functions

The two leading-twist T-odd structures are the Sivers and Boer-Mulders distributions. They are defined as

εijT pTiSTj

Mf⊥1T (x,p

2T) = −1

4Tr

[

(Φ(x,pT , S)− Φ(x,pT ,−S)) γ+]

+ h.c. , (80)

εijT pTj

Mh⊥1 (x,p

2T) =

1

4Tr

[

(Φ(x,pT , S) + Φ(x,pT ,−S)) iσi+ γ5]

+ h.c. . (81)

At tree level, these expressions vanish because there is no residual interaction between the active quark and the spec-tators; equivalently, there is no interference between two competing channels producing the complex amplitude whoseimaginary part gives the T-odd contribution. We can generate such structures by considering the interference betweenthe tree-level scattering amplitude and the single-gluon-exchange scattering amplitude in eikonal approximation, asshown in Fig. 3 (the Hermitean conjugate partner must also be considered). This corresponds just to the leading-twistone-gluon-exchange approximation of the gauge link operator of Eq. (5) [55].

10

PY

−l

p − P

p − l

Γ

FIG. 3: Interference between the one-gluon exchange diagram in eikonal approximation and the tree level diagram in thespectator model. The Hermitean conjugate diagram is not shown.

For the moment, we use Abelian gluons. The QCD color structure will be recovered at the end. The Feynman rulesto be used for the eikonal vertex and propagator are [1, 56]

ρ

= −iec nρ− ,

(−l)=

i

−l+ + iǫ, (82)

where ec is the color charge of the quark and the sign of iǫ for the eikonal line corresponds to the gauge link of SIDIS.In cut diagrams one must take the complex conjugate of these expressions for vertices and propagators on the rightof the final-state cut.The explicit form of the contribution Φ(1) to the correlation function corresponding to Fig. 3 is

Φ(1)(x,pT , S) ∼1

(2π)31

2(1− x)P+

(

M(0)(S)M(1)(S) +M(1)

(S)M(0)(S))∣

∣

∣

p2=τ(x,pT), (83)

where τ(x,pT) is defined in Eq. (8) and

M(1)(S) =

−∫

d4l

(2π)4iec Γs ρ n

ρ−( p/− l/+m)Ys U(P, S)

(D1 + iε) (D2 − iε) (D3 + iε) (D4 + iε)scalar diquark,

−∫

d4l

(2π)4iec ε

∗σ(P − p, λa) Γ

νσa ρ n

ρ−( p/− l/+m) dµν(p− l − P )Yµ

a U(P, S)

(D1 + iε) (D2 − iε) (D3 + iε) (D4 + iε)axial-vector diquark,

(84)where for convenience we have introduced the notation

D1 = l2 −m2g,

D2 = l+,

D3 = (p− l)2 −m2,

D4 = (P − p+ l)2 −M2X .

(85)

In order to explicitly calculate M(1), we need to model the gluon vertex with the scalar (Γs) and axial vector (Γa)diquark in Fig. 3:

Γρs = iec (2P − 2p+ l)ρ

Γνσa ρ = −iec

[

(2P − 2p+ l)ρ gνσ − (P − p+ (1 + κa)l)

σgνρ − (P − p− κal)ν gσρ

]

, (86)

where ec is the diquark color charge, which is the same for scalar and vector ones and identical to that of the quark; κais the diquark anomalous chromomagnetic moment. The structure of the vector diquark-gluon vertex resembles theone for the coupling between the photon and a spin-1 particle (see, e.g., Ref. [57]); for κa = 1 the standard point-likephoton-W coupling is recovered (see, e.g., Ref. [58]).The Sivers and Boer-Mulders functions can then be computed as

εijT pTiSTj

Mf⊥1T (x,p

2T) = −1

4

1

(2π)31

2(1− x)P+Tr

[(

M(1)(S)M(0)(S)−M(1)(−S)M(0)

(−S))

γ+]

+ h.c. , (87)

εijT pTj

Mh⊥1 (x,p

2T) =

1

4

1

(2π)31

2(1− x)P+Tr

[(

M(1)(S)M(0)(S) +M(1)(−S)M(0)

(−S))

iσi+ γ5

]

+ h.c. . (88)

11

Again, results have been produced for the three different choices of both Eq. (12) for the form factors at thenucleon-quark-diquark vertex, as well as of the axial-vector diquark propagator on each side of the diquark-gluonvertex in Fig. 3. Consistently with the case of T-even parton densities, here we show the results for the dipolar formfactor of Eq. (13) and for the light-cone transverse polarizations of the vector diquark, i.e. the first choice in Eq. (10),the other combinations being listed in App. B. Combining the rules (11) with the (86) ones, we can rewrite Eq. (87)and (88) as

f⊥ q(s)1T (x,p2

T) = −gs

4

1

(2π)3M e2c

2(1− x)P+

(1− x)2

[p2T+ L2

s(Λ2s)]

22 ImJs

1 (89)

f⊥ q(a)1T (x,p2

T) =

ga4

1

(2π)3M e2c

4(1− x)P+

(1− x)2

[p2T+ L2

a(Λ2a)]

22 ImJa

1 , (90)

h⊥ q(s)1 (x,p2

T) = f

⊥ q(s)1T (x,p2

T) (91)

h⊥ q(a)1 (x,p2

T) = − 1

xf⊥ q(a)1T (x,p2

T). (92)

Note that for scalar diquarks the spectator model gives the same result for the Sivers and the Boer-Mulders functions,independent of the choice of the nucleon-quark-diquark form factor (see App. B).In Eqs. (89) and ff., the expressions J1 contain the integral over the loop momentum, the denominators D1,2,3,4

defined in Eq. (85), and the evaluation of the trace of the projected amplitude. For instance (see App. B1)

Js1 =

∫

d4l

(2π)4gs(

(p− l)2)

(D1 + iε) (D2 − iε) (D3 + iε) (D4 + iε)4i(

l+ + 2(1− x)P+)(

l+M − P+(m+ xM)lT · pT

p2T

)

. (93)

To calculate its imaginary part, it is sufficient to make the replacements

1

D2 − iε→ 2πiδ(D2),

1

D4 + iε→ −2πiδ(D4) (94)

which corresponds to applying the Cutkosky rules [59], cutting the diquark propagator (D4) and the eikonalized quarkone (D2). We then get

2 Im Js1 =

∫

d4l

(2π)4gs(

(p− l)2)

D1D34(

l+ + 2(1− x)P+)

(


p2T

)

(2πi) δ(D2) (−2πi) δ(D4)

= −4P+ (m+ xM) (1− x) gs I1 .(95)

The calculation of I1 depends on the form factor used. Their calculation can be found in App. C. For the case of thedipolar form factor we obtain

− 4P+ (m+ xM) (1− x) gs Idip1 = gs

P+ (m+ xM) (1 − x)2

πL2s(Λ

2s) [p

2T+ L2

s(Λ2s)]

. (96)

If the T-odd structures were deduced from the Drell–Yan amplitude, the M(1) of Eq. (84) would involve a (l++ iε)propagator, leading to the opposite sign in the cutting rule for D2. In the spectator model, this is the origin of thepredicted sign change for f⊥

1T and h⊥1 when extracting them in Drell–Yan spin asymmetries rather than in SIDISones [16]. Analogously to Eq. (95), we obtain

2 Im Ja1 = −8P+ x (m+ xM) ga Idip

1 = ga2P+ x(1 − x) (m+ xM)

πL2a(Λ

2a) [p

2T+ L2

a(Λ2a)]

. (97)

By inserting these results in the model expressions of Eqs. (89) to (92), we come to the final form of the Sivers andBoer-Mulders functions with scalar and axial vector diquarks:

12

f⊥ q(s)1T (x,p2

T) = −g

2s

4

M e2c(2π)4

(1− x)3 (m+ xM)

L2s(Λ

2s) [p

2T+ L2

s(Λ2s)]

3(98)

f⊥ q(a)1T (x,p2

T) =

g2a4

M e2c(2π)4

(1− x)2 x (m+ xM)

L2a(Λ

2a) [p

2T+ L2

a(Λ2a)]

3(99)

h⊥ q(s)1 (x,p2

T) = f

⊥ q(s)1T (x,p2

T) (100)

h⊥ q(a)1 (x,p2

T) = − 1

xf⊥ q(a)1T (x,p2

T) . (101)

To connect the “Abelian” version of the gluon interaction to the QCD color interaction we shall apply the replace-ment [13]

e2c → 4πCFαs. (102)

The Sivers and Boer-Mulders functions obtained in our model behave as 1/p6Tat high p2

T, similarly to the f1 in

Eq. (44). As observed also in Ref. [60], this leads to a breaking of the positivity bounds [9] for sufficiently highvalues of p2

T. This problem is due to the fact that the T-odd functions have been calculated at order α1

S , while theT-even functions at order α0

S . At high p2T, QCD radiative corrections generate a 1/p2

Ttail for f1 and a 1/p4

Ttail for

f⊥1T [12]. Our model is supposed to be valid for p2

T∼ M2 and for reasonable choices of the parameters no problems

with positivity occurr in this region.Often the following transverse-momentum moments of the Sivers and Boer-Mulders functions are used:

f⊥ (1)1T (x) =

∫

dpT

p2T

2M2f⊥1T (x,p

2T)

f⊥ (1/2)1T (x) =

∫

dpT

|pT |2M

f⊥1T (x,p

2T) . (103)

In our model, they turn out to be

f⊥ q(s) (1)1T (x) = − g

2s

32

e2c(2π)3M

(m+ xM) (1 − x)3

[L2s(Λ

2s)]

2(104)

f⊥ q(a) (1)1T (x) =

g2a32

e2c(2π)3M

x (m+ xM) (1− x)2

[L2a(Λ

2a)]

2, (105)

h⊥ q(s) (1)1 (x) = f

⊥ q(s) (1)1T (x) (106)

h⊥ q(a) (1)1 (x) = − 1

xf⊥ q(a) (1)1T (x). (107)

f⊥ q(s) (1/2)1T (x) = − g2s

256

e2c(2π)2

(m+ xM) (1− x)3

[L2s(Λ

2s)]

5/2, (108)

f⊥ q(a) (1/2)1T (x) =

g2a256

e2c(2π)2

x (m+ xM) (1− x)2

[L2a(Λ

2a)]

5/2, (109)

h⊥ q(s) (1/2)1 (x) = f

⊥ q(s) (1/2)1T (x) (110)

h⊥ q(a) (1/2)1 (x) = − 1

xf⊥ q(a) (1/2)1T (x). (111)

E. T-odd functions: overlap representation

As already mentioned above, T-odd leading-twist parton distributions arise from the interference of two channelsleading to the same final state; for the case considered here (and depicted in Fig. 3), the two channels are given by

13

the tree-level and the single-gluon-exchange scattering amplitudes, respectively. In Ref. [61], it was suggested thatT-odd parton densities can also be represented by overlaps of LCWFs, as for their T-even partners, provided that asuitable operator is included to describe the final-state interactions (FSI) produced by the gluon rescattering. So far,this representation was fully developed in a spectator model only for the Sivers function with scalar diquarks [24, 26].Here, we generalize it to the case of axial-vector diquarks, as well as to the Boer-Mulders function. In this way, allleading-twist (T-even and T-odd) parton densities can be given by overlaps of LCWFs consistently within the model,contrary to the statement of Ref. [27].Following Ref. [26], for a nucleon transverse polarization state described by Eqs. (50) and (51) along a generic

direction ST = (cosφS , sinφS), and for an analogous quark state described by Eqs. (52) and (53) along SqT =(cosφSq

, sinφSq), we can rewrite the Sivers (80) and Boer-Mulders (81) functions according to the Trento Conven-

tions [62] (keeping in mind that λX is absent for the scalar diquark and λX = ± for the vector diquark) as

2 (ST × pT ) · PM

f⊥1T (x,p

2T) =

∫

dp′T

16π3G(x,pT ,p

′T)

∑

λq ,λX

[

ψ↑ ∗λqλX

(x,pT )ψ↑λqλX

(x,p′T)− ψ↓ ∗

λqλX(x,pT )ψ

↓λqλX

(x,p′T)]

+ h.c. , (112)

(SqT × pT ) · PM

h⊥1 (x,p2T) =

∫

dp′T

16π3G(x,pT ,p

′T)

1

2

∑

λN , λX

[

ψλN ∗↑λX

(x,pT )ψλN

↑λX(x,p′

T)− ψλN ∗

↓λX(x,pT )ψ

λN

↓λX(x,p′

T)]

+ h.c. . (113)

The above equations should be considered as assumptions, since it is not known a priori if the FSI operatorG(x,pT ,p

′T) can be isolated and is the same for all functions and all types of diquarks. In our model, it turns out to

be actually the same in all cases. In order to determine it, we must insert here above the expressions for the LCWFs ofSec. II B, and compare the results with the ones from Eqs. (89) to (92), after replacing ImJs

1 , ImJa1 with Eqs. (95-97),

respectively, while keeping the definition of Idip1 (see App. C). For the scalar diquark case, for example, we get

f⊥q(s)1T (x,p2

T) =

g2s8π3

M (1 − x)3 (m+ xM)

[p2T+ L2

s(Λ2s)]

2

∫

dp′T

ImG(x,pT ,p′T)

[p′ 2T

+ L2s(Λ

2s)]

2

(pT − p′T) · pT

p2T

. (114)

The above expression is identical to Eq. (89), after inserting Eq. (96) and the definition (C3) of Idip1 (with the harmless

substitution l′T↔ −l′

T), provided that

ImG(x,pT ,p′T) = − e2c

2 (2π)21

(pT − p′T)2

= −CFαs

2π

1

(pT − p′T)2, (115)

in agreement with the expression of Ref. [26]. Following similar steps, we recover the same result (115) also for theSivers function with axial-vector diquarks, and for the Boer-Mulders function as well. The FSI operator G(x,pT ,p

′T)

is indeed universal and describes a rescattering via one gluon-exchange, which corresponds to the expansion at firstorder of the gauge link operator of Eq. (5). Note that in other versions of the model (see App. B) the FSI cannot beas simple as Eq. (115), since we observe also a dependence on the vector diquark anomalous chromomagnetic momentκa, which is absent in the above equation. This does not imply that the FSI operator is not universal, but simplythat it could have additional parts that are not interacting with scalar diquarks and transversely polarized vectordiquarks.We close this section by observing that in our model we can generalize the relation between the first pT -moment of

the Sivers function and the nucleon anomalous magnetic moment κ, suggested in Ref. [26] in the simple scalar diquarkpicture. In fact, we define κ in terms of the Dirac form factor using the overlap representation for the nucleon matrixelement of the spin-flip electromagnetic current operator [51]:

κ =e

2MF2(0)

= − 1

qx − iqy

∑

k,n,λn

en

∫

dpT dx

16π3Ψ+ ∗

k (x,p′T, λn)Ψ

−k (x,pT , λn)

∣

∣

∣

∣

∣

p′

T=pT

, (116)

where the sum runs upon the number of Fock states k, the number of constituents n in each state k, and theirhelicities λn. Since in the diquark model of the nucleon initially at rest (PT = 0) there is only one Fock state with

14

two constituents, and the kinematics of the diquark is constrained to the one of the valence quark, the wave functionsΨ reduce to the usual LCWF [26]. The momentum conservation for the struck quark reads p′

T= pT + (1 − x) qT .

Distinguishing between κq(s) and κq(a) for scalar and axial-vector diquarks, respectively, Eq. (116) becomes

κq(s) = − 1

qx − iqy

∫

dpTdx

16π3

∑

λq

[

ψ+ ∗λq

(x,p′T)ψ−

λq(x,pT )

]

∣

∣

∣

∣

∣

p′

T=p

T

=g2s

(2π)2

∫ 1

0

dx1− x

12

(1 − x)3 (m+ xM)

[L2s(Λ

2s)]

3≡

∫ 1

0

dxκq(s)(x)

κq(a) = − 1

qx − iqy

∫

dpTdx

16π3

∑

λq,λa

[

ψ+ ∗λqλa

(x,p′T)ψ−

λqλa(x,pT )

]

∣

∣

∣

∣

∣

p′

T=p

T

= − g2a(2π)2

∫ 1

0

dxx

12

(1− x)3 (m+ xM)

[L2a(Λ

2a)]

3≡

∫ 1

0

dxκq(a)(x) . (117)

By comparison with the first pT -moment of f⊥ q(s)1T and f

⊥ q(a)1T in Eqs. (98) and (99), respectively,

f⊥ q(s)1T (x) =

∫

dpT f⊥ q(s)1T (x,p2

T)

= − g2s(2π)2

MCFαs (1− x)3 (m+ xM)

[2L2s(Λ

2s)]

3

f⊥ q(a)1T (x) =

∫

dpT f⊥ q(a)1T (x,p2

T)

=g2a

(2π)2MCFαs x (1 − x)2 (m+ xM)

[2L2a(Λ

2a)]

3, (118)

we deduce the relation

f⊥ q1T (x) = −3

2MCFαs

κq(x)

1− x, (119)

valid for both types of diquarks, from which we have

∫ 1

0

dx (1 − x) f⊥ q1T (x) = −3

2MCFαs κ

q (120)

that generalize the findings of Ref. [24, 26].

III. NUMERICAL RESULTS AND COMPARISON WITH VARIOUS PARAMETRIZATIONS

In this section, after fixing the parameters of the model by fitting some known distribution functions, we show thenumerical results of our model for a few selected TMDs.

A. Choice of model parameters

In order to fix the parameters of the model, we try to reproduce the parametrizations of parton distribution functionsextracted from experimental data. When doing this, however, we have to face the problem of choosing a scale Q2 atwhich our model can be compared to the parametrization. In principle, this scale should be considered as a furtherparameter of the model. However, we checked that the lowest possible value of Q2 is always preferred by the fit. Thisis not surprising, since probably the model is applicable to a very low scale, beyond the limit of applicability of theperturbative QCD evolution equations. Therefore, we have decided to compare it to a parametrization at the lowestpossible value of Q2.For the unpolarized distribution functions fu

1 and fd1 , we have chosen the parametrization of the ZEUS collabora-

tion [63] (ZEUS2002) at Q20 = 0.3 GeV2. This set of PDFs gives also an estimate of the errors, which is important to

perform a χ2 fit. Other parametrizations either do not reach such low Q2 or provide no error estimate.

15

For the helicity distributions gu1 and gd1 , we chose the leading-order (LO) version from Ref. [64] (GRSV2000) atQ2 = 0.26 GeV2. Since this parametrization comes with no error estimate, we assigned a fixed relative error of 10%and 25% to the up and down quark distributions, respectively, which is reasonably similar to the error estimates ofother parametrizations at higher Q2 (see, e.g., Ref. [65]).Finally, in order to perform the fit we arbitrarily chose to select from each parametrization 25 equally spaced points

in the range x = 0.1 to 0.7.The free parameters of the model include the quark mass m, the nucleon-quark-diquark coupling gX , the diquark

mass MX , and the cutoff ΛX in the nucleon-quark-diquark form factor, for X = s, a scalar and axial-vector diquarks.It turns out that in order to achieve a good fit we need also to make a distinction between the two isospin statesof the vector diquark. Hence, we will use ga Ma, and Λa, for the coupling, mass and cutoff of the vector isoscalardiquark with I3 = 0 (corresponding to the ud system), and g′a, M

′a, and Λ′

a, for the normalization, mass and cutoff ofthe vector isovector diquark with I3 = 1 (corresponding to the uu system).In order to reduce the number of free parameters, we decided to fix the value of the constituent quark mass to

m = 0.3 GeV. We checked that the results are not very sensitive to the value of this parameter.

To perform the fit, we need to discuss the relation between the functions fq(s)1 , f

q(a)1 and f

q(a′)1 , computed in the

model, and the functions fu1 and fd

1 of the global fits. For ease of interpretation, it is better to use normalized versions

of the fq(X)1 . Therefore, we write f

q(X)1 norm =

(

N2X/g

2X

)

fq(X)1 where NX are normalization constants determined by

imposing

π

∫ 1

0

dx

∫ ∞

0

dp2Tfq(X)1 norm(x,p

2T) = 1. (121)

Quite generally, the relation between quark flavors and diquark types can be written as

fu1 = c2s f

u(s)1 norm + c2a f

u(a)1 norm (122)

fd1 = c′2a f

d(a′)1 norm . (123)

We will refer to the coefficients cX as “couplings”, although they differ from the original couplings gX by the normal-ization constants NX . They are free parameters of the model.In past versions of the spectator diquark model [42], the quarks were assumed to occupy the lowest-energy available

orbital with positive parity (JP = 12

+); in this case, the proton wave function assumes an SU(4)=SU(2)⊗SU(2) spin-

isospin symmetry, leading to probabilistic weights 3:1:2 among the scalar isoscalar (quark u with diquark s), vectorisoscalar (quark u with diquark a), and vector isovector (quark d with diquark a′) configurations. Moreover, the overallsize of the couplings was adjusted to give a total number of three quarks. These choices led to the relations [42]

fu1 =

3

2fu(s)1 norm +

1

2fu(a)1 norm (124)

fd1 = f

d(a′)1 norm . (125)

There are two reasons to criticize this choice. First of all, in the present work the quark-diquark system can have anonvanishing relative orbital angular momentum, as shown in the previous section. Thus, the proton wave functionno longer displays an SU(4) symmetry. Secondly, strictly speaking the SU(4) decomposition gives coefficients thatare three times smaller then the ones in the above relation. This is because the total number of quarks “seen” in thespectator model is only one, since the other two are always hidden inside the diquark. This is actually a fundamentallimitation of the spectator model, it is independent of the SU(4) choice, and in our opinion it has not been sufficientlystressed in the literature. The only possible way out is to consider the diquark not as an elementary particle, but asformed by two quarks that can be also probed by the photon (see, e.g., Ref. [66]).A different way to see this problem is by considering the (longitudinal) momentum sum rule. Since also the diquarks

can carry momentum, they should be included in the corresponding sum rule.3 Using the handbag diagram of Fig. 2,

we calculated the corresponding diquark distribution function fX(q)1 for the active diquark in the state X and the

spectator quark with flavor q, again using the first choice in Eq. (10) (independently of the choice of form factor). Wefound the remarkable property

fX(q)1 (x) = f

q(X)1 (1 − x). (126)

3 A similar approach has been used in Ref. [25] to verify in the spectator model the validity of the so-called Burkardt sum rule [67], whichis related to transverse-momentum conservation.

16

By splitting the total proton momentum sum rule into the contributions of quarks, Pq, and of diquarks, PX , usingthe symmetry property (126) we get

Pq + PX =

∫ 1

0

dxx[

c2s fu(s)1 norm(x) + c2a f

u(a)1 norm(x) + c′2a f

d(a′)1 norm(x)

]

+

∫ 1

0

dxx[

c2s fs(u)1 norm(x) + c2a f

a(u)1 norm(x) + c′2a f

a′(d)1 norm(x)

]

=

∫ 1

0

dx[



d(a′)1 norm(x)

]

= c2s + c2a + c′2a .

(127)

It is therefore impossible in our spectator model to fulfill at the same time the momentum sum rule and the quarknumber sum rule.Although from the fundamental point of view it is more important to satisfy the momentum sum rule, from the

phenomenological point of view it is impossible to reproduce the parametrizations in a satisfactory way. We decidedtherefore to avoid imposing the momentum sum rule and let the fit choose the values of the parameters cX .

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.2

0.4

0.6

0.8

1.0

x

f1uHxL

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.1

0.2

0.3

0.4

0.5

x

f1dHxL

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.1

0.2

0.3

0.4

0.5

x

g1uHxL

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.20

-0.15

-0.10

-0.05

0.00

0.05

x

g1dHxL

FIG. 4: The distribution functions f1(x) (above) and g1(x) (below) for the up quark (left panel) and the down quark (rightpanel). Data are a selection of 25 equidistant points in 0.1 ≤ x ≤ 0.7 from the parametrizations of Ref. [63] (ZEUS2002)and Ref. [64] (GRSV2000) at LO, respectively (we assigned a constant relative error of 10% to gu1 and 25% to gd1 based oncomparisons with similar fits [65]). The curves represent the best fit (χ2/d.o.f. = 3.88) obtained with our spectator model. Thestatistical uncertainty bands correspond to ∆χ2 = 1.

In summary, we have 9 free parameters for the model. We fix them by fitting at the same time fu1 , f

d1 at Q2 = 0.3

GeV2 from Ref. [63], and gu1 , gd1 at Q2 = 0.26 GeV2 from Ref. [64] at LO. The fit was performed using the MINUIT

program. A χ2/d.o.f. = 3.88 was reached. The results are shown in Fig. 4. In spite of the very high χ2, the agreementis acceptable, except perhaps for the down quark helicity distribution. The error band is deduced from the covariancematrix given by MINUIT and represents the standard 1-σ statistical uncertainty (∆χ2 = 1). The correspondingvalues for the various model parameters are listed in Tab. I.

17

Diquark MX (GeV) ΛX (GeV) cX

Scalar s (uu) 0.822 ± 0.053 0.609 ± 0.038 0.847 ± 0.111

Axial-vector a (ud) 1.492 ± 0.173 0.716 ± 0.074 1.061 ± 0.085

Axial-vector a′ (uu) 0.890 ± 0.008 0.376 ± 0.005 0.880 ± 0.008

TABLE I: Results for the model parameters with dipolar nucleon-quark-diquark form factor and light-cone transverse polariza-tions of the vector diquark: the diquark masses MX , the cutoffs ΛX in the form factors, and the cX couplings for X = s, a, a′

scalar isoscalar, vector isoscalar, and vector isovector diquarks. The fit was performed using the MINUIT program on theparametrization of f1(x) from Ref. [63] (ZEUS2002), and of g1(x) from Ref. [64] (GRSV2000) at LO, reaching a χ2/d.o.f. =3.88.

0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

pT2 HGeV2L

f1u Hx, pT

2 L

x=0.02x=0.2

x=0.5

0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

pT2 HGeV2L

f1d Hx, pT

2 L

x=0.02

x=0.2

x=0.5

FIG. 5: The p2T dependence of the unpolarized distribution f1(x,p

2T ) for up (left panel) and down quark (right panel). Different

lines correspond to different values of x. The downturn of the function fu1 at relatively small x is due to wavefunctions with

nonzero orbital angular momentum.

B. Unpolarized parton densities

With the above model parameters, the proton momentum fraction Pq carried by valence quarks, is

Pq =

∫ 1

0

dxx[



d(a′)1 norm(x)

]

=

∫ 1

0

dxx[

fu1 (x) + fd

1 (x)]

≈ 0.584± 0.010 , (128)

which is consistent with the ZEUS result of 0.55 [63].While for fd

1 only the vector-isovector diquark plays a role, for fu1 it turns out that the contributions from the scalar

and vector diquark have about the same size. The vector diquark is always dominant at high x. However, we knowthat the model is not reliable in the limit x → 1. In fact, the behavior at high x does not follow the predictions ofRef. [68], since our model does not correctly take into account the dominant dynamics in that region.We consider now the p2

Tdependence of the unpolarized distribution function obtained in our model. In Fig. 5 we

show the behavior of the up and down components as functions of p2Tfor a few values of the variable x.

First of all, we observe that fu1 displays a nonmonotonic behavior at x ≤ 0.02. This is due to the contribution

from LCWFs with nonzero orbital angular momentum. Although the details of where and how this feature occursis model-dependent, it is generally true that the contribution of LCWFs with one unit of orbital angular momentumfalls linearly with p2

Tfor p2

T→ 0. This behavior is sharply different from the contribution of LCWFs with no orbital

angular momentum. This simple example shows how the study of the p2Tdependence of unpolarized TMDs can

therefore already expose some effects due to orbital angular momentum.Finally, we observe that in our model the average quark transverse momentum decreases as x increases, and that

down quarks on average carry less transverse momentum than up quarks. Although this is just a model result, ageneral message can be derived: the widely used assumption of a flavor-independent quark transverse momentumdistribution is already falsified in a relatively simple model (see also Ref. [69]).

18

C. Longitudinally polarized parton densities

The model parameters of Tab. I produce the axial charge

gA =

∫ 1

0

dx[

gu1 (x)− gd1(x)]

= 0.966± 0.038 , (129)

in excellent agreement with the value 0.969± 0.096 deduced from the GRSV parametrization [64].It is, however, evident from Fig. 4 that our description of the down quark helicity distribution is in bad disagreement

with the GRSV parametrization at large x. Nevertheless, we point out that there is a qualitative agreement with theparametrization of the so-called BBS model of Ref. [70] and the analogous parametrization of Ref. [68]. In particular,our model shows the same feature highlighted in this latter reference, namely that the contribution of the LCWFswith nonvanishing orbital angular momentum is dominant at high x. This is true in all distribution functions, butbecomes particularly evident for the down helicity distribution, since the contribution from the LCWFs ψ+

++ and ψ++−

(carrying nonzero orbital angular momentum) are positive and make the distribution positive at x > 0.5.

0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

pT2 HGeV2L

up, x=0.02

f1u

-g1u

f1u

+g1u

0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

pT2 HGeV2L

down, x=0.02

f1d

-g1d

f1d

+g1d

FIG. 6: The p2T dependence of the distributions f1(x,p

2T )− g1(x,p

2T ) (solid line) and f1(x,p

2T ) + g1(x,p

2T ) (dashed line) for up

(left panel) and down quark (right panel), at x = 0.02. The difference in their behavior is due to the different role played inthe two combinations by wavefunctions with nonzero orbital angular momentum.

The effect of orbital angular momentum becomes even more evident when considering the p2Tbehavior of the helicity

distribution function. As an illustration, we show in Fig. 6 the behavior of the combinations f1(x,p2T) − g1(x,p

2T)

and f1(x,p2T) + g1(x,p

2T). In the case of the scalar diquark, LCWFs with one unit of orbital angular momentum

are filtered by the first combination. In the case of the vector diquark, the situation is opposite. The down quarkdistribution is entirely given by the vector diquark, therefore the f1(x,p

2T)+ g1(x,p

2T) sum clearly turns down to zero

for p2T→ 0. For the up quark, the situation is less clear due to the simultaneous presence of scalar and vector diquark

contributions. However, at x = 0.02 the vector diquark is responsible for the nontrivial shape of f1(x,p2T)+ g1(x,p

2T).

It is interesting also to investigate the p2Tbehavior of gu1 alone, shown in Fig. 7. There is a dramatic change of

behavior for different values of x, due to the difference between the scalar and vector diquark components of thefunction. If the spectator is a scalar diquark, for pT = 0, where the LCWFs with orbital angular momentum vanish,

the spin of the up quark has to be parallel to that of the proton, thus gu(s)1 (x, 0) ≥ 0. At high transverse momentum,

where LCWFs with Lz = 1 dominate, the spin of the up quark has to be antiparallel to that of the proton, thus

gu(s)1 (x,∞) ≤ 0. The situation is exactly reversed in the case of the vector diquark. As is already visible in Eqs. (58)and (59), at high transverse momentum the vector diquark always dominates and gives a positive result. At lowtransverse momentum, the relative size of the functions L2

X(Λ2X) in the denominator determines which contribution is

dominant. At higher x the scalar diquark dominates and gives a positive gu1 (x, 0), while at lower x the vector diquarkdominates and gives a negative gu1 (x, 0).Once again, apart from the details specific to our model, these examples show that the exploration of the p2

T

dependence of the unpolarized and helicity distribution functions can expose very interesting features of the innerstructure of the nucleon, related in particular to orbital angular momentum.

19

0.1 0.2 0.3 0.4 0.5 0.6-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

pT2 HGeV2L

g1u Hx, pT

2 L

x=0.02

x=0.2

FIG. 7: The p2T dependence of the helicity distribution gu1 (x,p

2T ). Different lines correspond to different values of x.

0.2 0.4 0.6 0.8 1.0-0.1

0.0

0.1

0.2

0.3

0.4

x

x h1u HxL

Q2=0.3 GeV

Q2=2.5 GeV

0.2 0.4 0.6 0.8 1.0-0.15

-0.10

-0.05

0.00

0.05

x

x h1d HxL

Q2=0.3 GeV

Q2=2.5 GeV

FIG. 8: The transversity distribution xh1(x) for up (left panel) and down quark (right panel). Dashed (solid) line for the modelresult before (after) the evolution at LO using the code of Ref. [71] up to the scale of the parametrization from Ref. [72], whoseuncertainty band due to errors in the fit parameter is represented by the shaded area.

D. Transversity

In Fig. 8, the predictions of the spectator diquark model for the transversity distribution are compared with theonly available parametrization of Ref. [72]. In the left panel, xhu1 (x) is shown, whereas xhd1(x) is shown in the rightpanel. All the model results at the assumed original scale Q2

0 = 0.3 GeV2 are represented by the dashed line. Thesolid line indicates the result after applying the DGLAP evolution at LO up to the scale Q2 = 2.5 GeV2 using the codefrom Ref. [71]. The latter scale pertains the parametrization of Ref. [72], whose errors in the fit parameters producethe uncertainty band represented by the shaded area. The model is in reasonable agreement with the parametrization,with the maxima in the correct position and a somewhat too small result for the up quark at small x. It should alsobe kept in mind that the present data reach at most x ≈ 0.4 [73, 74] and, moreover, the ansatz of Ref. [72] does notallow for a sign change.4

Interestingly, for the up quark the model predicts a change of sign at x ∼ 0.5. To our knowledge, no other model oftransversity displays this feature (see Ref. [76] and references therein; see also recent calculations in Refs. [66, 77, 78]).

4 We point out that new fits of the transversity distribution functions have been presented at some conferences [75] but not published yet.

20

The reason for this sign change is that the contribution of the vector diquark is negative, as evident from Eq.(79). Inour model, at moderate x the scalar diquark contribution is dominant, whereas at sufficiently high x the contributionof the vector diquark becomes in absolute size bigger, thus leading to the sign change. Other versions of the diquarkmodel, even with vector diquarks, may not show this property. This is already evident from inspecting the results(listed in the appendices) for different choices of the diquark polarization sum. We don’t think that our modelcalculation should be trusted more than others. Nevertheless, it might be interesting to contemplate the possibilityof a sign change when choosing a form for the parametrization of the transversity function in “global fits.”

0.2 0.4 0.6 0.8 1.0-0.05

0.00

0.05

0.10

0.15

0.20

0.25

ÈpT È HGeVL

x h1u Hx,pT

2 LÈx=0.1

0.2 0.4 0.6 0.8 1.0-0.15

-0.10

-0.05

0.00

0.05

ÈpT È HGeVL

x h1d Hx,pT

2 LÈx=0.1

FIG. 9: Same as in the previous figure, but for the pT dependence of transversity at x = 0.1.

In Fig. 9, the same comparison is performed as in the previous figure, but for the pT dependence of the transversityat x = 0.1, as it is deduced from Eqs. (68,69). Again, there is a reasonable agreement between model predictionsand parametrizations but for the trend of the result for the up quark at |pT | > 0.3 GeV/c. However, we stress thatthe comparison may be affected by the different scale of the model results (Q2 = 0.3 GeV2) and the one at whichthe parametrizations are extracted (Q2 = 2.5 GeV2). The proper evolution of the TMDs has not been consideredyet. It is interesting to point out that in our model hu1 (x,p

2T) changes sign at pT ∼ 0.5 GeV. This is due to the fact

that the vector diquark contribution is always negative and dominant at high pT . For the down quark, we note thathd1(x, 0) = 0, because the vector-diquark contribution to h1 is entirely given by LCWFs with nonvanishing orbitalangular momentum.

0.2 0.4 0.6 0.8 1.0-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

x

x f1 T⊥ H1L u HxL

0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05

0.06

x

x f1 T⊥ H1L d HxL

FIG. 10: The first pT -moment xf⊥ (1)1T (x) of the Sivers function; left (right) panel for up (down) quark. Solid line for the

results of the spectator diquark model. Darker shaded area for the uncertainty band due to the statistical error of the quarkparametrizations from Ref. [79], lighter one from Ref. [80].

21

E. Sivers function

In Fig. 10, the xf⊥ (1)1T (x) moment of the Sivers function, predicted using Eqs. (104) and (105), is given by the

solid line and it is compared with two different parametrizations of the same observable. The darker shaded arearepresents the uncertainty due to the statistical errors in the parametrization of Ref. [79], while the lighter shadedarea corresponds to the same for Ref. [80]. Left panel refers to the up quark, right panel to down quark. Firstof all, we observe the agreement between the signs of the various flavor components, which also agree with thefindings from calculations on the lattice [81]. Also the maxima are reached at approximately the same x ∼ 0.3 as theparametrizations. Instead, the “strength” of the asymmetry (related to the modulus of each moment) is too muchweaker for the down quark, while it seems reasonable for the up one. Again, it must be stressed that no evolutionwas applied in the displayed model results.

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

pX HGeVL

pYHG

eVL

fu�p Hx=0.1L

ΡHG

eV-

2L

4

0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

pX HGeVL

pYHG

eVL

fd�p Hx=0.1L

ΡHG

eV-

2L

12

0

FIG. 11: The model result for the spin density of unpolarized quarks in transversely polarized protons (see text for the precisedefinition) in pT space at x = 0.1. Left panel for up quark, right panel for down quark. The circle with the arrow indicatesthe direction of the proton polarization.

According to the Trento conventions [62], we define the spin density of unpolarized quarks with flavor q in trans-versely polarized protons as

fq/p↑(x,pT ) = f q1 (x,p

2T)− f⊥ q

1T (x,p2T)(P × pT ) · S

M. (130)

In a SIDIS experiment, typically P is antialigned to the z axis that points in the direction of the momentumtransfer q. Hence, if the proton polarization is chosen along the x axis, the spin density (130) shows an asymmetry inmomentum space along the py direction, whose size is driven by the Sivers function. In Fig. 11, we show fq/p↑(0.1,pT )for q = u (left panel) and q = d (right panel). Since the Sivers function for the up (down) quark is negative (positive),the density is deformed towards positive (negative) values of py. This feature is in agreement with the lattice resultsof Ref. [81] and with the signs of the anomalous magnetic moments κq [61].

F. Boer-Mulders function

In Fig. 12, the xh⊥ (1)1 (x) and xh

⊥ (1/2)1 (x) moments of the Boer-Mulders function, as deduced from Eqs. (106,107)

and (110,111), are displayed in the left and right panel, respectively. The solid lines correspond to the results forthe up quark; dashed lines for the down quark. For the Boer-Mulders function, the only available parametrizationappeared recently in [22], but the overall normalization depends on a parameter ω that cannot be fixed with availableexperimental information. Our result agrees in sign and shape with that extraction. The absolute values of ourfunctions correspond to ω ≈ 0.3. We remark that there is full agreement between the sign of the u and d componentsand the aforementioned lattice calculations [81], as observed also in a different version of the spectator model [36] and

22

0.2 0.4 0.6 0.8 1.0-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

x

x h1⊥ H1L HxL

up

down

0.2 0.4 0.6 0.8 1.0-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

x

x h1⊥ H1�2L HxL

up

down

FIG. 12: The xh⊥ (1)1 (x) (left) and xh

⊥ (1/2)1 (x) (right) moments of the Boer-Mulders function. Solid and dashed lines for up

and down quarks, respectively.

in the bag model [47]. This agreement seems to be a general feature, as argued in Ref. [29]. 5

In Fig. 13, we show, again at x = 0.1, the spin density of transversely polarized quarks with flavor q in unpolarizedprotons, related to the Boer-Mulders effect by [62]

fq↑/p(x,pT ) =1

2

[

f q1 (x,p

2T)− h⊥ q

1 (x,p2T)(P × pT ) · Sq

M

]

, (131)

where now the quark polarization Sq points along x. Since the Boer-Mulders function is negative for both flavors(see Fig. 12), the related spin density is always deformed towards positive py, again in agreement with the latticeresults [81].

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

pX HGeVL

pYHG

eVL

fu�p Hx=0.1L

ΡHG

eV-

2L

2

0

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

pX HGeVL

pYHG

eVL

fd�p Hx=0.1L

ΡHG

eV-

2L

6

0

FIG. 13: The model result for the spin density of transversely polarized quarks in unpolarized protons (see text for the precisedefinition) in pT space at x = 0.1. Left panel for up quark, right panel for down quark. The arrow inside the circle indicatesthe direction of the quark polarization.

5 A different result for the sign of the down quark Boer-Mulders function was obtained in Ref. [46], probably due to a mistake in thatcalculation (see App. B 4).

23

IV. CONCLUSIONS

We have presented a systematic calculation of all leading-twist parton distributions in the nucleon in a diquarkspectator model. We have generated the relative phase necessary to produce T-odd structures by approximating thegauge link operator with a one gluon-exchange interaction. All results have been presented in analytic form andinterpreted in terms of overlaps of light-cone wavefunctions.We tried to extend and improve the spectator model calculations presented in Refs. [36, 42, 46] by considering

several choices of the axial-vector diquark polarization states and of the nucleon-quark-diquark form factor. We listedthe analytic expressions for all possible choices in the appendices. We critically reconsidered some of the limits of themodel and the choice of model parameters used in the past literature. In particular, we showed that the spectatordiquark model is not able to reproduce both the quark number and momentum sum rule at the same time, becausethe diquark is considered as a charged parton, hence active in the sum rules. We argued that the proton wave functiondoes not show the usual SU(4)=SU(2)⊗SU(2) symmetry [42], since the quark-diquark system in its ground state canhave a nonvanishing relative orbital angular momentum.For numerical studies, we chose the version of the model that in our opinion is more sensible and practical, i.e., the

one where only light-cone transverse polarizations of the diquark are present and a dipolar form factor is used. Weidentified nine free parameters of the model and we fixed them by reproducing the phenomenological parametrizationof unpolarized [63] and longitudinally polarized [64] parton distributions at the lowest available scale, i.e. Q2 = 0.3and 0.26 GeV2, respectively.Whenever possible, results have been compared with available parametrizations. For the chiral-odd transversity

distribution, there is only one available from Ref. [72], which was deduced from SSA data at Q2 = 2.5 GeV2. ThepT -integrated model result, once evolved to this scale using the code from Ref. [71] at LO, displays a satisfactory

overall agreement. The f⊥ (1)1T (x) moment of the chiral-even T-odd Sivers function f⊥

1T was compared with theparametrizations of Refs. [79, 80]. There is agreement between the signs of the various flavor components andbetween the positions of the maxima in x, but the absolute value of the function is somewhat too small for the downquark. The comparison is affected by the difference of the scales, since evolution equations for the Sivers function

have not been used. We also plotted the h⊥ (1)1 (x) and h

⊥ (1/2)1 (x) moments of the chiral-odd T-odd Boer-Mulders

function h⊥1 . We have also shown the quark spin densities defined in the Trento conventions [62], as produced in turnby the Sivers or the Boer-Mulders effects. For unpolarized quarks in transversely polarized protons, the spin densityfq/p↑ is linked to f⊥

1T , while for transversely polarized quarks in unpolarized protons the fq↑/p is linked to h⊥1 . Fortransverse polarizations along the x axis, the contour plot in the quark momentum space of such densities at x = 0.1displays a distortion in the py direction, whose sign is consistent with the lattice findings for the corresponding spindensities in impact parameter space [81].Using the model parton densities discussed above, various spin, beam, and azimuthal asymmetries in semi-inclusive

hadronic reactions can be predicted, which are of interest for several experimental collaborations. Model calculationscan be useful to interpret experimental measurements, helping us to explore spin-orbit parton correlations insidehadrons and shed light on the well-known puzzle of the proton spin sum rule.

Acknowledgments

F. C. and M. R. would like to thank B. Pasquini for useful discussions.This work is part of the European Integrated Infrastructure Initiative in Hadronic Physics project under Contract

No. RII3-CT-2004-506078.Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S.

Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscriptfor U.S. Government purposes.

APPENDIX A: T-EVEN FUNCTIONS IN DIFFERENT VARIATIONS OF THE MODEL

In this Appendix we list the leading-twist T-even parton densities obtained in the context of our spectator diquarkmodel, for all the choices of axial-vector diquark polarization sum and nucleon-quark-diquark vertex. To avoidoverloading the notation, we will use the same ones for the parameters involved (gX , MX , ΛX). However, it must bekept in mind that the numerical value of these parameters can be different in the various versions of the model.

24

1. Scalar diquark

The results for the scalar diquark are

fq(s)1 (x,p2

T) =

g2s(2π)3

[(m+ xM)2 + p2T] (1− x)

2 [p2T+ L2

s(m2)]2

(A1)

gq(s)1L (x,p2

T) =

g2s(2π)3

[(m+ xM)2 − p2T] (1− x)

2 [p2T+ L2

s(m2)]2

, (A2)

gq(s)1T (x,p2

T) =

g2s(2π)3

M (m+ xM) (1 − x)

[p2T+ L2

s(m2)]2

, (A3)

h⊥ q(s)1L (x,p2

T) = −gq(s)1T (x,p2

T) , (A4)

hq(s)1T (x,p2

T) = f

q(s)1 (x,p2

T) , (A5)

h⊥ q(s)1T (x,p2

T) = − g2s

(2π)3M2 (1− x)

[p2T+ L2

s(m2)]2

, (A6)

where we recall that M is the nucleon mass and m is the mass of the parton. From the latter two densities, weconstruct the contribution of the scalar diquark to the transversity:

hq(s)1 (x,p2

T) = h

q(s)1T (x,p2

T) +

p2T

2M2h⊥ q(s)1T (x,p2

T)

=g2s

(2π)3(m+ xM)2 (1 − x)

2 [p2T+ L2

s(m2)]2

=1

2

(

fq(s)1 (x,p2

T) + g

q(s)1 (x,p2

T))

.

(A7)

The above results are valid for a point-like nucleon-quark-diquark coupling. For the other form factors it is sufficientto apply the replacements

g2s → g2s(1 − x)2[p2

T+ L2

s(m2)]2

[p2T+ L2

s(Λ2s)]

4dipolar form factor, (A8)

g2s → g2se−[p2

T+L2X(m2)]/[(1−x) Λ2

X ] exponential form factor. (A9)

The integrated results are obviously different for the three form-factor choices. In all cases the transversity functionsaturates the Soffer bound, i.e.,

hq(s)1 (x) =

1

2

(

fq(s)1 (x) + g

q(s)1 (x)

)

. (A10)

• Point-like coupling (to avoid divergences we assume that the p2Tintegration is extended up to a finite cutoff Λ2

s)

fq(s)1 (x) =

g2s (1− x)

(2π)2

(m+ xM)2 Λ2s − L2

s(m2) Λ2

s + L2s(m

2) [Λ2s + L2

s(m2)] log

(

Λ2s

L2s(m

2) + 1)

4L2s(m

2) [Λ2s + L2

s(m2)]

, (A11)

gq(s)1 (x) =

g2s (1− x)

(2π)2

(m+ xM)2 Λ2s + L2

s(m2) Λ2

s − L2s(m

2) [Λ2s + L2

s(m2)] log

(

Λ2s

L2s(m

2) + 1)

4L2s(m

2) [Λ2s + L2

s(m2)]

, (A12)

• Dipolar form factor [same as Eqs. (74) and (76)]

fq(s)1 (x) =

g2s(2π)2

[2 (m+ xM)2 + L2s(Λ

2s)] (1− x)3

24L6s(Λ

2s)

(A13)

gq(s)1 (x) =

g2s(2π)2

[2 (m+ xM)2 − L2s(Λ

2s)] (1− x)3

24L6s(Λ

2s)

(A14)

25

• Exponential form factor

fq(s)1 (x) =

g2s(2π)2

1

4

{

e−2L2s(m

2)/[(1−x)Λ2s]

1− x

L2s(m

2)

[

(m+ xM)2 − L2s(m

2)]

− Γ

(

0,2L2

s(m2)

(1− x) Λ2s

)

2[

(m+ xM)2 − L2s(m

2)]

− (1− x) Λ2s

Λ2s

}

,

(A15)

gq(s)1 (x) =

g2s(2π)2

1

4

{

e−2L2s(m

2)/[(1−x)Λ2s]

1− x

L2s(m

2)

[

(m+ xM)2 + L2s(m

2)]

− Γ

(

0,2L2

s(m2)

(1− x) Λ2s

)

2 [(m+ xM)2 + L2s(m

2)] + Λ2s (1− x)

Λ2s

}

,

(A16)

where Γ is the incomplete Γ function, defined as

Γ(a, z) =

∫ ∞

z

ta−1 e−tdt . (A17)

2. Axial-vector diquark with light-cone transverse polarization only

The unintegrated parton densities are

fq(a)1 (x,p2

T) =

g2a(2π)3

p2T(1 + x2) + (m+ xM)2 (1 − x)2

2 [p2T+ L2

a(m2)]2 (1 − x)

gq(a)1L (x,p2

T) =

g2a(2π)3

p2T(1 + x2)− (m+ xM)2 (1 − x)2

2 [p2T+ L2

a(m2)]2 (1 − x)

,

gq(a)1T (x,p2

T) =

g2a(2π)3

M x (m+ xM)

[p2T+ L2

a(m2)]2

,

h⊥ q(a)1L (x,p2

T) = g

q(a)1T (x,p2

T)/x ,

hq(a)1T (x,p2

T) = − g2a

(2π)3xp2

T

[p2T+ L2

a(m2)]2 (1 − x)

,

h⊥ q(a)1T (x,p2

T) = 0 ,

hq(a)1 (x,p2

T) = h

q(a)1T (x,p2

T) . (A18)

The above results are valid for a point-like nucleon-quark-diquark coupling. For the other form factors it is sufficientto apply the replacements in Eqs. (A8) and (A9).The integrated results are


a)

fq(a)1 (x) =

g2a(2π)2

1

4L2a(m

2) [Λ2a + L2

a(m2)] (1 − x)

[

xΛ2a

[

(M2 −m2) (1− x2) + 2mM (1 − x)2 −M2a (1 + x2)

]

+ L2a(m

2) [Λ2a + L2

a(m2)] (1 + x2) log

(

Λ2a

L2a(m

2)+ 1

)

]

,

gq(a)1 (x) =

g2a(2π)2

1

4L2a(m

2) [Λ2a + L2

a(m2)] (1 − x)

{

L2a(m

2) [Λ2a + L2

a(m2)] (1 + x2) log

(

Λ2a

L2a(m

2)+ 1

)

− Λ2a

[

(1− x)mM [x(1− x) (2M −m) + 2m] + x(1− x)M3 (x − 2x2 − 1) + x(1 + x2)MM2a

]

}

,

26

hq(a)1 (x) = − g2a

(2π)2

x[

Λ2a[(1− x) (xM2 −m2)− xM2

a ] + L2a(m

2) [Λ2a + L2

a(m2)] log

(

Λ2a

L2a(m

2) + 1)]

2L2a(m

2) [Λ2a + L2

a(m2)] (1− x)

. (A19)

• Dipolar form factor [same as Eqs. (75,77, 79)]

fq(a)1 (x) =

g2a(2π)2

[2 (m+ xM)2 (1− x)2 + (1 + x2)L2a(Λ

2a)] (1− x)

24L6a(Λ

2a)

,

gq(a)1 (x) = − g2a

(2π)2[2 (m+ xM)2 (1− x)2 − (1 + x2)L2

a(Λ2a)] (1 − x)

24L6a(Λ

2a)

,

hq(a)1 (x) = − g2a

(2π)2x(1 − x)

12L4a(Λ

2a). (A20)


fq(a)1 (x) =

g2a(2π)2

1

4Λ2a L

2a(m

2) (1− x)2

{

Λ2a L

2a(m

2) (1− x) (1 + x2) Γ

(

0, 2L2a(m

2)

(1 − x) Λ2a

)

+[

(m+ xM)2 (1− x)2 − L2a(m

2) (1 + x2)]

×[

(1− x) Λ2a e

−2L2a(m

2)/[(1−x) Λ2a] − 2L2

a(m2) Γ

(

0, 2L2a(m

2)

(1− x) Λ2a

)]

}

,

gq(a)1 (x) =

g2a(2π)2

1

4Λ2a L

2a(m

2) (1− x)2

{

Λ2a L

2a(m

2) (1− x) (1 + x2) Γ

(

0, 2L2a(m

2)

(1 − x) Λ2a

)

+[

(m+ xM)2 (1− x)2 + L2a(m

2) (1 + x2)]

×[

2L2a(m

2) Γ

(

0, 2L2a(m

2)

(1 − x) Λ2a

)

− (1 − x) Λ2a e

−2L2a(m

2)/[(1−x) Λ2a]

]

}

,

hq(a)1 (x) =

g2a(2π)2

x

2Λ2a (1− x)2

{

Λ2a (1 − x)

[

e−2L2a(m

2)/[(1−x) Λ2a] − Γ

(

0, 2L2a(m

2)

(1− x) Λ2a

)]

− 2L2a(m

2) Γ

(

0, 2L2a(m

2)

(1 − x) Λ2a

)

}

. (A21)

3. Axial-vector diquark including also longitudinal polarization


fq(a)1 (x,p2

T) =

g2a(2π)3

1

4 [p2T+ L2

a(m2)]2M2

a (1− x)

[

p4T+ xM2

a (2p2T+ xM2

a )

+ (1 − x)2[

p2T(M2 +m2 + 2M2

a) + 2m2M2a + 6xmMM2

a + 2x2M2M2a +m2M2 (1− x)2

]

]

,

gq(a)1L (x,p2

T) =

g2a(2π)3

1

4 [p2T+ L2

a(m2)]2M2

a (1− x)

[

p4T+ xM2

a (2p2T+ xM2

a )

+ (1 − x)2[

p2T(2M2

a −m2 −M2 − 4mM)− 2m2M2a − 2xmMM2

a − 2x2M2M2a +m2M2 (1 − x)2

]

]

,

gq(a)1T (x,p2

T) =

g2aM

(2π)3(m+M)p2

T−mM (m+M) (1− x)2 + xM2

a [M(2x− 1) +m]

2 [p2T+ L2

a(m2)]2M2

a

,

27

h⊥ q(a)1L (x,p2

T) =

g2aM

(2π)3(m+M)p2

T−mM (m+M) (1− x)2 + xMM2

a −mM2a (x− 2)

2 [p2T+ L2

a(m2)]2M2

a

,

hq(a)1T (x,p2

T) = − g2a

(2π)3p4

T+ [2xM2

a + (m2 +M2) (1− x)2]p2T+ [xM2

a +mM (1− x)2]2

4 [p2T+ L2

a(m2)]2M2

a (1 − x),

h⊥ q(a)1T (x,p2

T) =

g2a(2π)3

M2 (m+M)2 (1 − x)

2 [p2T+ L2

a(m2)]2M2

a

,

hq(a)1 (x,p2

T) = − g2a

(2π)3p4

T+ [2xM2

a − 2mM (1− x)2]p2T+ [xM2

a +mM (1− x)2]2

4M2a (1− x) [p2

T+ xM2

a + (1− x) (m2 − xM2)]2. (A22)

The above results are valid for a point-like nucleon-quark-diquark coupling. For the other form factors it is sufficientto apply the replacements in Eqs. (A8) and (A9).The integrated results are


a)

fq(a)1 (x) =

g2a(2π)2

1

8L2a(m

2) [Λ2a + L2

a(m2)]M2

a (1− x)

[

Λ2a [L

4a(m

2) + Λ2a L

2a(m

2)]

+ (1 − x)2[

2M2a [m

2 − L2a(m

2)] + x [(m2 −M2)2 −M2a (m

2 − 6mM +M2)] + x2 2M2M2a

]

+ L2a(m

2) [Λ2a + L2

a(m2)] (1− x) log

(

Λ2a

L2a(m

2)+ 1

)

[(1 + x) (m2 −M2)− 2M2a (1− x)]

]

,

gq(a)1 (x) =

g2a(2π)2

1

8L2a(m

2) [Λ2a + L2

a(m2)]MM2

a x (1 − x)

{

xML2a(m

2) Λ2a [Λ

2a + L2

a(m2)] + Λ2

a (1− x)2

×[

L2a(m

2)[

mM2 (2x− 1) +m2M (x− 2)−m3 + xM (M2 − 2M2a)]

+m5 (1 − x)−m4M (x − 2) +m3M2ax+m3M2 (1− x2) +m2M3x (x2 − 2x− 1)

+mM2M2ax+mM4x (2x2 − x− 1) + x3M3 (M2 − 2M2

a)

]

+ (1 − x)L2a(m

2) [Λ2a + L2

a(m2)] log

(

Λ2a

L2a(m

2)+ 1

)

×[

L2a(m

2) (m+M)−m3 (1− x) +m2M (x2 − 2x− 1)− xm (3M2 (1− x) +M2a )

+ xM (M2a (1 − 2x) + 2xM2)

]}

,

hq(a)1 (x) = − g2a

(2π)21

8L2a(m

2) [Λ2a + L2

a(m2)]M2

a (1 − x)

[

L2a(m

2) Λ2a (L

2a(m

2) + Λ2a)

+ (1 − x)2 Λ2a [(m+M)2 L2

a(m2) + x (m−M)2 ((m+M)2 −M2

a )]

− 2L2a(m

2) (L2a(m

2) + Λ2a) (m+M) (m− xM) (1 − x) log

(

Λ2a

L2a(m

2)+ 1

)

]

. (A23)

• Dipolar form factor

fq(a)1 (x) =

g2a(2π)2

1− x

48L6a(Λ

2a)M

2a

[

2L4a(Λ

2a) + 2xM2

a (L2a(Λ

2a) + xM2

a )

+ (1− x)2[

(2M2a +m2 +M2)L2

a(Λ2a) + 4m2M2

a + x 12mMM2a + x2 4M2M2

a + 2m2M2 (1 − x)2]

]

,

28

gq(a)1 (x) =

g2a(2π)2

1− x

48L6a(Λ

2a)M

2a

[

2L4a(Λ

2a) + 2xM2

a (L2a(Λ

2a) + xM2

a )

+ (1− x)2[

(2M2a −m2 −M2 − 4mM)L2

a(Λ2a)− 4m2M2

a − x 4mMM2a − x2 4M2M2

a

+2m2M2 (1 − x)2]

]

,

hq(a)1 (x) = − g2a

(2π)21− x

24L6a(Λ

2a)M

2a

[

L4a(Λ

2a) + [xM2

a −mM (1− x)2]L2a(Λ

2a) + [xM2

a +mM (1 − x)2]2]

. (A24)


fq(a)1 (x) =

g2a(2π)2

1

16L2a(m

2) Λ2aM

2a (1 − x)2

×{

2

[

Λ2a (1− x) e−2 [p2

T+L2a(m

2)]/[(1−x)Λ2a] − 2L2

a(m2) Γ

(

0,2L2

a(m2)

(1 − x) Λ2a

)]

×[

(1− x)2[

m2 (2M2a +M2 (1− x)2)− L2

a(m2) (2M2

a +m2 +M2) + 2xMM2a (xM + 3m)

]

+ (1− x)xM2a (xM2 −m2) + L2

a(m2) (L2

a(m2)− xM2

a )

]

+ Λ2a L

2a(m

2) (1 − x)

[

(1− x)2 2Γ

(

0,2L2

a(m2)

(1− x) Λ2a

)

(m2 +M2 + 2M2a)

+ (1− x) Λ2a e

−2 [p2T+L2

a(m2)]/[(1−x)Λ2

a] + 4Γ

(

0,2L2

a(m2)

(1− x) Λ2a

)

[xM2a − L2

a(m2)]

]}

gq(a)1 (x) =

g2a(2π)2

1

16L2a(m

2) Λ2aM

2a (1 − x)2

×{

[

Λ2a (1− x) e−2 [p2

T+L2a(m

2)]/[(1−x)Λ2a] − 2L2

a(m2) Γ

(

0,2L2

a(m2)

(1− x) Λ2a

)]

×[

(1− x)4 2m2M2 − (1− x)2[

2mM [xM2a − L2

a(m2)]− L2

a(m2) (m+M)2

]

+ (1− x)[

Λ2a L

2a(m

2) + 2M2a (m2 (x− 2) + (2x− 1)x2M2)

]

+ 2L2a(m

2)[

L2a(m

2)− xM2a − 2M2

a (1 − x)2]

]

− 2L2a(m

2) Λ2a (1− x) Γ

(

0,2L2

a(m2)

(1− x) Λ2a

)

[

L2a(m

2)− 2xM2a + (1− x)2 [(m+M)2 + 2mM − 2M2

a ]]

}

,

hq(a)1 (x) =

g2a(2π)2

1

16L2a(m

2) Λ2aM

2a (1 − x)2

{[

Λ2a (1 − x)

[

sinh

(

2p2

T+ L2

a(m2)

(1 − x) Λ2a

)

− cosh

(

2p2

T+ L2

a(m2)

(1− x) Λ2a

)]

+ 2L2a(m

2) Γ

(

0,2L2

a(m2)

(1 − x) Λ2a

)

] [

(1− x)4 2m2M2 + (1− x)2 4mM [xM2a + L2

a(m2)]

+ (1− x)[

Λ2a L

2a(m

2) + 2xM2a (xM2 −m2)

]

+ 2L2a(m

2) (L2a(m

2)− xM2a )

]

+ 2L2a(m

2) Λ2a (1− x) Γ

(

0,2L2

a(m2)

(1− x) Λ2a

)

[

L2a(m

2)− 2xM2a + (1− x)2 2mM

]

}

. (A25)

29

4. Axial-vector diquark including also time-like polarization


fq(a)1 (x,p2

T) =

g2a(2π)3

[p2T+ (m+ xM)2 + 2mMx] (1− x)

2 [p2T+ L2

a(m2)]2

gq(a)1L (x,p2

T) = − g2a

(2π)3[−p2

T+m2 + x2M2] (1− x)

2 [p2T+ L2

a(m2)]2

,

gq(a)1T (x,p2

T) = − g2a

(2π)3M2 x(1− x)

[p2T+ L2

a(m2)]2

,

h⊥ q(a)1L (x,p2

T) =

g2a(2π)3

mM (1− x)

[p2T+ L2

a(m2)]2

,

hq(a)1T (x,p2

T) = −xh⊥ q(a)

1L (x,p2T) ,

h⊥ q(a)1T (x,p2

T) = 0 ,

hq(a)1 (x,p2

T) ≡ h

q(a)1T (x,p2

T) . (A26)

The above results are valid for a point-like nucleon-quark-diquark coupling. For the other form factors it is sufficientto apply the replacements in Eqs. (A8) and (A9). The result for f1 with dipolar form factor corresponds to thatobtained in Ref. [46].The integrated results are


a)

fq(a)1 (x) =

g2a (1− x)

(2π)2

[(m+M)2 + 2mM −M2a ]xΛ

2a + L2

a(m2) [Λ2

a + L2a(m

2)] log(

Λ2a

L2a(m

2) + 1)

4L2a(m

2) [Λ2a + L2

a(m2)]

,

gq(a)1 (x) = −g

2a (1− x)

(2π)2

Λ2a (L

2a(m

2) +m2 + x2M2)− L2a(m

2) [Λ2a + L2

a(m2)] log

(

Λ2a

L2a(m

2) + 1)

4L2a(m

2) [Λ2a + L2

a(m2)]

,

hq(a)1 (x) = −g

2a (1− x) Λ2

a

(2π)2xmM

2L2a(m

2) [Λ2a + L2

a(m2)]

. (A27)


fq(a)1 (x) =

g2a(2π)2

[

L2a(Λ

2a) + 2 [(m+ xM)2 + 2xmM ]

]

(1 − x)3

24L6a(Λ

2a)

,

gq(a)1 (x) =

g2a(2π)2

[L2a(Λ

2a)− 2 (m2 + x2M2)] (1− x)3

24L6a(Λ

2a)

,

hq(a)1 (x) = − g2a

(2π)2mM x (1 − x)3

6L6a(Λ

2a)

. (A28)


fq(a)1 (x) =

g2a(2π)2

1

4

{

e−2L2a(m

2)/[(1−x) Λ2a]

[(m+ xM)2 + 2mxM − L2a(m

2)] (1− x)

L2a(m

2)

− Γ

(

0, 2L2a(m

2)

(1− x) Λ2a

)

2 [(m+ xM)2 + 2mxM − L2a(m

2)]− (1− x) Λ2a

Λ2a

}

,

gq(a)1 (x) =

g2a(2π)2

1

4

{

Γ

(

0, 2L2a(m

2)

(1− x) Λ2a

)

2 [m2 + x2M2 + L2a(m

2)] + (1 − x) Λ2a

Λ2a

30

− e−2L2a(m

2)/[(1−x)Λ2a]

(1− x) [m2 + x2M2 + L2a(m

2)]

L2a(m

2)

}

,

hq(a)1 (x) =

g2a(2π)2

mxM

2

{

2

Λ2a

Γ

(

0,2L2

a(m2)

(1− x) Λ2a

)

− e−2L2a(m

2)/[(1−x)Λ2a]

1− x

L2a(m

2)

}

. (A29)

APPENDIX B: T-ODD FUNCTIONS IN DIFFERENT VARIATIONS OF THE MODEL

As a continuation of App. A, here we list the Sivers and Boer-Mulders functions, namely the leading-twist T-oddparton densities obtained in the context of our spectator diquark model, again for all the combinations of diquarkpropagators and nucleon-quark-diquark vertices.

1. Scalar diquark

For scalar diquarks, we have

f⊥ q(s)1T (x,p2

T) = −gs(p

2)

4

1

(2π)3M e2c

2(1− x)P+

2 ImJs1

p2 −m2

h⊥ q(s)1 (x,p2

T) = f

⊥ q(s)1T (x,p2

T) , (B1)

where the Js1 integral is defined as

(

εijTpTiSTj

)

Js1 =

∫

d4l

(2π)4gs(

(p− l)2)

(D1 + iε) (D2 − iε) (D3 + iε) (D4 + iε)

Tr

[

( p/− l/+m) ( P/+M) γ5 S/ ( p/+m) (2P − 2p+ l)ρ nρ− γ+

]

=

∫

d4l

(2π)4gs(

(p− l)2)

(D1 + iε) (D2 − iε) (D3 + iε) (D4 + iε)4i(

l+ + 2(1− x)P+)

(

l+M εijTpTiSTj − P+(m+ xM) εij

TlTiSTj

)

,

with D1, D2, D3, D4 defined in Eq. (85). The imaginary part of Js1 can be extracted by using the Cutkosky cutting

rules on the loop diagram of Fig. 3, which in the present case amount to put on shell the eikonalized virtual quarkpropagatorD2 and the virtual scalar diquark propagatorD4. The resulting δ functions (see below) reduce the integralin Eq. (B2) to a bidimensional integral in d2lT . In general, for a n-dimensional integral

∫

dnl lρf(l, p) the term lρ canbe replaced by the expression pρ(l · p)/p2. For the present case n = 2 and with the identification lρ = lTi, pρ = pTi,we finally can write

2 Im Js1 =

∫

d4l

(2π)4gs(

(p− l)2)

D1D34(

l+ + 2(1− x)P+)

(


p2T

)

(2πi) δ(D2) (−2πi) δ(D4)

= −4P+ (m+ xM) (1 − x) gs I1 .(B2)

The explicit expression of I1 clearly depends on the choice of the nucleon-quark-scalar diquark vertex form factor.The Boer-Mulders calculation gives exactly the same results as the Sivers one, in the scalar diquark framework,

since the relevant trace over Dirac-Lorentz structures is now given by

Tr[

( p/− l/+m) (P/+M) ( p/+m) (2P−2p+l)ρ nρ−iσ

i+γ5

]

= −4i(

l++2(1−x)P+)(

l+M εijTpTj−P+(m+xM) εij

TlTj

)

.

(B3)

• Point-like coupling

2 ImJs1 = gs

∫

d4l

(2π)41

D1D34(

l+ + 2(1− x)P+)

(


p2T

)

(2πi) δ(D2) (−2πi) δ(D4)

= −4P+ (m+ xM) (1− x) gs Ip.l.1 = −gs

P+ (m+ xM) (1− x)

πp2T

log

(

L2s(m

2) + p2T

L2s(m

2)

)

,

(B4)

31

where Ip.l.1 is calculated in App. C.

Using Eq. (8), the final result is then

f⊥ q(s)1T (x,p2

T) = −g

2s

4

M e2c(2π)4

(m+ xM) (1− x)

p2T[L2

s(m2) + p2

T)]

log

(

L2s(m

2) + p2T

L2s(m

2)

)

h⊥ q(s)1 (x,p2

T) = f

⊥ q(s)1T (x,p2

T) . (B5)

• Dipolar form factor. The final results, already given in Eq. (98) and (100), are

f⊥ q(s)1T (x,p2

T) = −g

2s

4

M e2c(2π)4

(1− x)3 (m+ xM)

L2s(Λ

2s) [p

2T+ L2

s(Λ2s)]

3

h⊥ q(s)1 (x,p2

T) = f

⊥ q(s)1T (x,p2

T) (B6)


2 ImJs1 = gs

∫

d4l

(2π)4e[(p−l)2−m2]/Λ2

s

D1D34(

l+ + 2(1− x)P+)

(


p2T

)

(2πi) δ(D2) (−2πi) δ(D4)

= −4P+ (m+ xM) (1− x) gs Iexp1

= −gsP+ (m+ xM) (1− x)

πp2T

[

Γ

(

0,L2s(m

2)

(1− x) Λ2s

)

− Γ

(

0,L2s(m

2) + p2T

(1 − x) Λ2s

)]

,

and Iexp1 is calculated in App. C.

The final results are then

f⊥ q(s)1T (x,p2

T) = −g

2s

4

M e2c(2π)4

(m+ xM) (1 − x)

p2T[L2

s(m2) + p2

T]e−[p2

T+L2s(m

2)]/[(1−x)Λ2s]

×[

Γ

(

0,L2s(m

2)

(1− x) Λ2s

)

− Γ

(

0,L2s(m

2) + p2T

(1 − x) Λ2s

)]

,

h⊥ q(s)1 (x,p2

T) = f

⊥ q(s)1T (x,p2

T) . (B7)

2. Axial-vector diquark with light-cone transverse polarization only

We have

f⊥ q(a)1T (x,p2

T) =

ga(p2)

4

1

(2π)3M e2c

4(1− x)P+

2 ImJa1

p2 −m2

h⊥ q(a)1 (x,p2

T) =

ga(p2)

4

1

(2π)3M e2c

4(1− x)P+

2 ImJ′(a)1

p2 −m2, (B8)

where now the Ja1 and J ′ a

1 integrals are defined as

(

εijTpTiSTj

)

Ja1 =

∫

d4l

(2π)4ga(

(p− l)2)

(D1 + iε) (D2 − iε) (D3 + iε) (D4 + iε)

Tr

[

( p/− l/+m) γµ γ5 ( P/+M) γ5 S/ γα γ5 ( p/+m) dµν(p− l − P ) dσα(P − p)

[

(2P − 2p+ l)ρ gνσ − (P − p+ (1 + κa)l)


]

nρ− γ+

]

(B9)

and

(

−εijTpTj

)

J ′ a1 =

∫

d4l

(2π)4ga(

(p− l)2)

(D1 + iε) (D2 − iε) (D3 + iε) (D4 + iε)

Tr

[

( p/− l/+m) γµ γ5 (P/ +M) γα γ5 ( p/+m) dµν(p− l − P ) dσα(P − p)

[

(2P − 2p+ l)ρ gνσ − (P − p+ (1 + κa)l)


]

nρ− iσi+γ5

]

, (B10)

32

and the explicit expressions of the dµν(p − l − P ) and dσα(P − p) structures are those expressed in the first line inEq. (10).


a)

2 Im Ja1 = −8P+x (m+ xM) ga Ip.l.

1 = −ga2P+ x (m+ xM)

πp2T

log

(

L2a(m

2) + p2T

L2a(m

2)

)

2 ImJ ′ a1 = 8P+ (m+ xM) ga Ip.l.

1 = ga2P+ (m+ xM)

πp2T

log

(

L2a(m

2) + p2T

L2a(m

2)

)

, (B11)

where Ip.l.1 is the same integral as in Eq. (B4) but with the substitution Ls(m

2) ↔ La(m2).

Using again Eq. (8), the final result is

f⊥ q(a)1T (x,p2

T) =

g2a4

M e2c(2π)4

x (m+ xM)

p2T[L2

a(m2) + p2

T]log

(

L2a(m

2) + p2T

L2a(m

2)

)

h⊥ q(a)1 (x,p2

T) = − 1

xf⊥ q(a)1T (x,p2

T) . (B12)

• Dipolar form factor. The final results, already given in Eq. (99) and (101), are

f⊥ q(a)1T (x,p2

T) =

g2a4

M e2c(2π)4

(1− x)2 x (m+ xM)

L2a(Λ

2a) [p

2T+ L2

a(Λ2a)]

3

h⊥ q(a)1 (x,p2

T) = − 1

xf⊥ q(a)1T (x,p2

T) . (B13)


2 ImJa1 = −8P+x (m+ xM) ga Iexp

1

= −ga2P+ x (m+ xM)

πp2T

[

Γ

(

0,L2a(m

2)

(1 − x) Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1− x) Λ2a

)]

,

2 Im J ′ a1 = 8P+ (m+ xM) ga Iexp

1

= ga2P+ (m+ xM)

πp2T

[

Γ

(

0,L2a(m

2)

(1− x) Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1− x) Λ2a

)]

. (B14)

The final result is, then,

f⊥ q(a)1T (x,p2

T) =

g2a4

M e2c(2π)4

x (m+ xM)

p2T[L2

a(m2) + p2

T]e−[p2

T+L2a(m

2)]/[(1−x) Λ2a]

×[

Γ

(

0,L2a(m

2)

(1− x) Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1 − x) Λ2a

)]

,

h⊥ q(a)1 (x,p2

T) = − 1

xf⊥ q(a)1T (x,p2

T) . (B15)

3. Axial-vector diquark including also longitudinal polarization

We have

f⊥ q(a)1T (x,p2

T) =

ga(p2)

4

1

(2π)3M e2c

4(1− x)P+

2 Im Ja1

p2 −m2

h⊥ q(a)1 (x,p2

T) =

ga(p2)

4

1

(2π)3M e2c

4(1− x)P+

2 Im J ′ a1

p2 −m2, (B16)

where the Ja1 and J ′ a

1 integrals are defined as in Eqs. (B9) and (B10), respectively, but now the second line in Eq. (10)is employed for the dµν(p− l− P ) and dσα(P − p) Lorentz structures.

33


a)

2 ImJa1 = ga

∫

d4l

(2π)41

D1D3(2πi) δ(D2) (−2πi) δ(D4)

×{

X(x,p2T)

(

1 +l2T

2M2a

)

lT ·(

pT − 1

2lT

)

+

(

lT · pT

p2T

) [

Y1(x,p2T) + Y2(x,p

2T) (lT · pT ) + Y3(x,p

2T) l2

T

]

+

W (x,p2T)

(

lT · ST

S2T

)

ST · [lT − 2pT ]

}

,

2 ImJ ′ a1 = ga

∫

d4l

(2π)41

D1D3(2πi) δ(D2) (−2πi) δ(D4)

×{

−X(x,p2T)

(

1 +l2T

2M2a

)

lT ·(

pT − 1

2lT

)

+

(

lT · pT

p2T

)

[

Y1(x,p2T) − Y2(x,p

2T) (lT · pT )− Y3(x,p

2T) l2

T+

+ 2M2a [(m+ xM) (1− x)− xm] +M2

a [m (κa − 1)−M(1 + κa)x] l2T

]}

, (B17)

where the integrals X,W, Yi, i = 1− 3 are listed in App. C. Unfortunately, most of the above combinations aredivergent under the dlT integration. This is a typical pathology when choosing the point-like form factor forthe nucleon-quark-diquark vertex, without any ad-hoc cut-off.


2 ImJa1 = ga

∫

d4l

(2π)41

D1 (D3)2(2πi) δ(D2) (−2πi) δ(D4)

×{

X(x,p2T)

(

1− 1

2l2T+

lT · pT

p2T

l2T

2M2a

− l4T

4M4a

)

+

W (x,p2T)

(

lT · ST

S2T

)

ST · [lT − 2pT ] +

(

lT · pT

p2T

)[

Y1(x,p2T) + Y2(x,p

2T)lT · pT

p2T

+ Y3(x,p2T) l2

T

]

}

,

2 ImJ ′ a1 = ga

∫

d4l

(2π)41

D1 (D3)2(2πi) δ(D2) (−2πi) δ(D4)

×{

−X(x,p2T)

(

1 +l2T

2M2a

)

lT ·(

pT − 1

2lT

)

+

(

lT · pT

p2T

)

[

Y1(x,p2T)− Y2(x,p

2T) lT · pT − Y3(x,p

2T) l2

T+

2M2a [(m+ xM) (1− x) −mx] +M2

a [m (κa − 1)−M(1 + κa)x] l2T

]}

, (B18)

where the integrals X,W, Yi, i = 1− 3 are listed in App. C. Unfortunately, most of the above combinations aredivergent under the dlT integration, unless a dipolar form factor is considered with a higher degree, for exampleproportional to [p2

T+L2

a(Λ2a)]

−3 in Eq. (13). This would introduce a 1/(D3)3 term inside Eq. (B18), instead of

34

1/(D3)2, and grant the convergence of the various integrals. With this very choice, we obtain

2 ImJa1 = ga

(1− x)2

2P+

{

X(x,p2T)

[

p2TI ′ dip1 +

(

p2T

2M2a

− 1

2

)

I ′ dip2 − 1

4M2a

I ′ dip5

]

+

Y1(x,p2T) I ′ dip

1 + Y2(x,p2T) I ′ dip

3 + Y3(x,p2T) I ′ dip

2 +

W (x,p2T)(

I ′ dip7 − 2pT · STI ′ dip

6

)

}

,

2 Im J ′ a1 = ga

(1− x)2

2P+

{

X(x,p2T)

[

−p2TI ′ dip1 +

(

1

2− p2

T

2M2a

)

I ′ dip2 +

1

4M2a

I ′ dip5

]

+

Y1(x,p2T) I ′ dip

1 − Y2(x,p2T) I ′ dip

3 − Y3(x,p2T) I ′ dip

2 +

2M2a [(m+ xM) (1 − x)−mx] I ′ dip

1 +M2a [m (κa − 1)−M(1 + κa)x] I ′ dip

2

}

, (B19)

where the integrals I ′ dipi , i = 1− 7, are listed in App. C.


f⊥ q(a)1T (x,p2

T) = − g

2a

32

M e2c(2π)3

(1− x)4

(P+)2 [L2a(Λ

2a) + p2

T]3

{

X(x,p2T)

[

p2TI ′ dip1 +

(

p2T

2M2a

− 1

2

)

I ′ dip2 − 1

4M2a

I ′ dip5

]

+ Y1(x,p2T) I ′ dip

1 + Y2(x,p2T) I ′ dip

3 + Y3(x,p2T) I ′ dip

2 +W (x,p2T)(

I ′ dip7 − 2pT · ST I ′ dip

6

)

}

,

h⊥ q(a)1 (x,p2

T) = − g

2a

32

M e2c(2π)3

(1− x)4

(P+)2 [L2a(Λ

2a) + p2

T]3

{

X(x,p2T)

[

−p2TI ′ dip1 +

(

1

2− p2

T

2M2a

)

I ′ dip2 +

1

4M2a

I ′ dip5

]

+ Y1(x,p2T) I ′ dip

1 − Y2(x,p2T) I ′ dip

3 − Y3(x,p2T) I ′ dip

2 + 2M2a [(m+ xM) (1 − x)−mx] I ′ dip

1

+M2a [m (κa − 1)−M(1 + κa)x] I ′ dip

2

}

. (B20)


2 ImJa1 = ga

∫

d4l

(2π)4e[(p−l)2−m2]/Λ2

a

D1D3(2πi) δ(D2) (−2πi) δ(D4)

×{

X(x,p2T)

(

1 +l2T

2M2a

)

lT ·(

pT − 1

2lT

)

+

W (x,p2T)

(

lT · ST

S2T

)

ST · [lT − 2pT ] +

(

lT · pT

p2T

)[

Y1(x,p2T) + lT · pT Y2(x,p

2T) + l2

TY3(x,p

2T)

]

}

=1

2P+

{

X(x,p2T)

[

p2TIexp1 +

(

p2T

2M2a

− 1

2

)

Iexp2 − 1

4M2a

Iexp5

]

+

Y1(x,p2T) Iexp

1 + Y2(x,p2T) Iexp

3 + Y3(x,p2T) Iexp

2 +

W (x,p2T) (Iexp

7 − 2pT · ST Iexp6 )

}

,

2 ImJ ′ a1 = ga

∫

d4l

(2π)4e[(p−l)2−m2]/Λ2

a

D1D3(2πi) δ(D2) (−2πi) δ(D4)

35

×{

−X(x,p2T)

(

1 +l2T

2M2a

)

lT ·(

pT − 1

2lT

)

+

(

lT · pT

p2T

)

[

Y1(x,p2T)− lT · pT Y2(x,p

2T)− l2

TY3(x,p

2T)+

+ 2M2a [(m+ xM) (1− x)−mx] +M2

a [m(κa − 1)−M(1 + κa)x] l2T

]}

=1

2P+

{

X(x,p2T)

[

−p2TIexp1 +

(

1

2− p2

T

2M2a

)

Iexp2 +

1

4M2a

Iexp5

]

+

Y1(x,p2T) Iexp

1 − Y2(x,p2T) Iexp


2 +

2M2a [(m+ xM) (1− x)−mx] Iexp

1 +M2a [m(κa − 1)−M(1 + κa)x] Iexp

2

}

, (B21)

where the integrals X,W, Yi, i = 1− 3, and Iexpi , i = 1− 7, are listed in App. C.


f⊥ q(a)1T (x,p2

T) = − g

2a

32

M e2c(2π)3

e−[p2T+L2

a(m2)]/[(1−x)Λ2

a]

(P+)2 [L2a(m

2) + p2T]

{

X(x,p2T)

[

p2TIexp1 +

(

p2T

2M2a

− 1

2

)

Iexp2 − 1

4M2a

Iexp5

]

+ Y1(x,p2T) Iexp

1 + Y2(x,p2T) Iexp

3 + Y3(x,p2T) Iexp

2 +W (x,p2T) (Iexp

7 − 2pT · ST Iexp6 )

}

,

h⊥ q(a)1 (x,p2

T) = − g

2a

32

M e2c(2π)3

e−[p2T+L2

a(m2)]/[(1−x)Λ2

a]

(P+)2 [L2a(m

2) + p2T]

{

X(x,p2T)

[

−p2TIexp1 +

(

1

2− p2

T

2M2a

)

Iexp2 +

1

4M2a

Iexp5

]

+ Y1(x,p2T) Iexp



2 + 2M2a [(m+ xM) (1 − x)−mx] Iexp

1

+M2a [m(κa − 1)−M(1 + κa)x] Iexp

2

}

. (B22)

4. Axial-vector diquark including also time-like polarization

We have

f⊥ q(a)1T (x,p2

T) =

ga(p2)

4

1

(2π)3M e2c

4(1− x)P+

2 Im Ja1

p2 −m2

h⊥ q(a)1 (x,p2

T) =

ga(p2)

4

1

(2π)3M e2c

4(1− x)P+

2 Im J ′ a1

p2 −m2, (B23)

where the Ja1 and J ′ a

1 integrals are defined as in Eqs. (B9) and (B10), respectively, but now the last line in Eq. (10))is employed for the dµν(p− l− P ) and dσα(P − p) Lorentz structures.


a)

2 ImJa1 = −4P+x [m (2κa + 1) +M (2κa x+ 1)] ga Ip.l.

1

= −gaP+ x [m (2κa + 1) +M (2κa x+ 1)]

πp2T

log

(

L2a(m

2) + p2T

L2a(m

2)

)

2 ImJ ′ a1 = 4P+ [m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]] ga Ip.l.

1

= gaP+ [m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]]

πp2T

log

(

L2a(m

2) + p2T

L2a(m

2)

)

, (B24)

where Ip.l.1 is the same integral as in Eq. (B4) but with the substitution Ls(m

2) ↔ La(m2).

36

Using again Eq. (8), the final result is

f⊥ q(a)1T (x,p2

T) =

g2a8

M e2c(2π)4

x [m (2κa + 1) +M (2κa x+ 1)]

p2T[L2

a(m2) + p2

T]

log

(

L2a(m

2) + p2T

L2a(m

2)

)

h⊥ q(a)1 (x,p2

T) = −g

2a

8

M e2c(2π)4

m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]

p2T[L2

a(m2) + p2

T]

log

(

L2a(m

2) + p2T

L2a(m

2)

)

. (B25)


2 ImJa1 = 4P+x (1 − x) [m (2κa + 1) +M (2κa x+ 1)] ga Idip

1

= gaP+ x (1 − x) [m (2κa + 1) +M (2κa x+ 1)]

π L2a(Λ

2a) [L

2a(Λ

2a) + p2

T]

2 ImJ ′ a1 = −4P+ (1− x) [m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]] ga Idip

1

= −gaP+ (1− x) [m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]]

π L2a(Λ

2a) [L

2a(Λ

2a) + p2

T]

. (B26)

The final result is

f⊥ q(a)1T (x,p2

T) =

g2a8

M e2c(2π)4

x (1 − x)2 [m (2κa + 1) +M (2κa x+ 1)]

L2a(Λ

2a) [L

2a(Λ

2a) + p2

T]3

,

h⊥ q(a)1 (x,p2

T) = −g

2a

8

M e2c(2π)4

(1− x)2 [m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]]

L2a(Λ

2a) [L

2a(Λ

2a) + p2

T]3

. (B27)

We find a discrepancy between these results and those of Eqs. (18) and (24) in Ref. [46], probably due to errorsin that calculation.


2 ImJa1 = −4P+x [m (2κa + 1) +M (2κa x+ 1)] ga Iexp

1

= −gaP+ x [m (2κa + 1) +M (2κa x+ 1)]

πp2T

[

Γ

(

0,L2a(m

2)

(1− x) Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1− x) Λ2a

)]

,

2 ImJ ′ a1 = 4P+ [m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]] ga Iexp

1

= gaP+ [m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]]

πp2T

[

Γ

(

0,L2a(m

2)

(1− x) Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1− x) Λ2a

)]

.

(B28)


f⊥ q(a)1T (x,p2

T) =

g2a8

M e2c(2π)4

x [m (2κa + 1) +M (2κa x+ 1)]

p2T[L2

a(m2) + p2

T]

e−[p2T+L2

a(m2)]/[(1−x) Λ2

a]

×[

Γ

(

0,L2a(m

2)

(1 − x) Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1− x) Λ2a

)]

,

h⊥ q(a)1 (x,p2

T) = −g

2a

8

M e2c(2π)4

m [(2κa − 1)x+ 2] + xM [(κa − 1) 2x+ 3]

p2T[L2

a(m2) + p2

T]

e−[p2T+L2

a(m2)]/[(1−x) Λ2

a]

×[

Γ

(

0,L2a(m

2)

(1 − x) Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1− x) Λ2a

)]

. (B29)

APPENDIX C: USEFUL INTEGRALS

In this appendix, we calculate the relevant integrals that repeatedly show up in the expressions of T-odd partondensities for all choices of nucleon-quark-diquark form factors, when vector diquark propagators are represented inthe first and last forms of Eq. (10).

37

We will systematically use the substitutions l′T= lT − pT and y = l′ 2

T+ L2

X(m2), or y = l′ 2T

+ L2X(Λ2

X) for thedipolar form factor, where LX is defined in Eq. (8) for X = s, a scalar and axial-vector diquarks, respectively. Wewill also encounter the following angular integrals, where θ is defined as the angle between l′

Tand pT , and φ, φS , are

the azimuthal angles of pT and ST with respect to the scattering plane:

∫ 2π

0

dθ|l′

T||pT | cos θ + p2

T

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

= π (1− sgn(|l′T| − |pT |)) ,

∫ 2π

0

dθ

[

|l′T||pT | cos θ + p2

T

]2

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=π

2

(

−l′ 2T

+ 3p2T+∣

∣l′ 2T

− p2T

∣

∣

)

,

∫ 2π

0

dθ|l′

T||ST | cos[θ + (φ− φS)] + |pT ||ST | cos(φ − φS)

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

= π|ST ||pT |

cos(φ − φS) (sgn(|pT | − |l′T|) + 1) ,

∫ 2π

0

dθ(|l′

T||ST | cos[θ + (φ− φS)] + |pT ||ST | cos(φ− φS))

2

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

= πS2

T

2p2T

[

(

p2T− l′ 2

T+ |p2

T− l′ 2

T|)

cos 2(φ− φS)

+ 2p2T

]

. (C1)

• Point-like coupling

Ip.l.1 =

∫

dl′T

(2π)2(l′

T+ pT ) · pT

p2T

1

(l′T+ pT )2 [l′ 2T + L2

X(m2)]

=1

(2π)2p2T

∫ ∞

0

d|l′T||l′

T| 1

l′ 2T

+ L2X(m2)

∫ 2π

0

dθ|l′

T||pT | cos θ + p2

T

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=1

(2π)p2T

∫ |pT |

0

d|l′T||l′

T| 1

l′ 2T

+ L2X(m2)

=1

4πp2T

∫ L2X(m2)+p2

T

L2X(m2)

dy

y=

1

4πp2T

log

(

L2X(m2) + p2

T

L2X(m2)

)

; (C2)


Idip1 = −

∫

dl′T

(2π)2(l′

T+ pT ) · pT

p2T

(1− x)

(l′T+ pT )2 [l′ 2T + L2

X(Λ2X)]2

= − (1− x)

4πp2T

∫ L2X(Λ2

X )+p2T

L2X(Λ2

X)

dy

y2= − (1− x)

4πL2X(Λ2

X) [L2X(Λ2

X) + p2T]; (C3)


Iexp1 =

∫

dl′T

(2π)2(l′

T+ pT ) · pT

p2T

e−[l′ 2T +L2X(m2)]/[(1−x)Λ2

X ]

(l′T+ pT )2 [l′ 2T + L2

X(m2)]

=1

(2π)p2T

∫ |pT |

0

d|l′T||l′

T| e

−[l′ 2T +L2X(m2)]/[(1−x)Λ2

X ]

l′ 2T

+ L2X(m2)

=1

(4π)p2T

∫ L2X(m2)+p2

T

L2X(m2)

dy

ye−y/[(1−x)Λ2

X ]

=1

4πp2T

[

Γ

(

0,L2X(m2)

(1− x)Λ2X

)

− Γ

(

0,L2X(m2) + p2

T

(1− x)Λ2X

)]

. (C4)

Next, we list the coefficients and calculate the relevant integrals that are needed to construct T-odd parton densitiesfor all choices of nucleon-quark-diquark form factors, when vector diquarks are represented in the second form inEq. (10).

38

X = −16(m+M) (P+)2 κaM2

a

,

W = −8(m+M) (P+)2 (κa − 1) (1− x)

M2a

,

Y1 = −8(P+)2

M2a

[

κa

(

2m3 − 3xm3 +Mx2m2 + 2Mm2 − 3Mxm2 −M2xm+ 2M2axm+M3x2 −M3x

− [p2T+ L2

a(m2)] (M +m)

)

+ (1 + κa)M2ax [M(1 + x) +m] + [p2

T+ L2

a(m2)]m (κa − 1) (x− 1)

+m (κa − 1)x2 (M2a +m2) +m3 (2x− 1) +M2m (1− 2x+ (1 + κa)x

2)

]

,

Y2 = −8m (P+)2 (κa − 1) (1− x)

M2a

,

Y3 =4(P+)2

M4a

[

κa

(

−m3 + xm3 −Mx2m2 − 2Mm2 + 3Mxm2 −M2m− 2M2x2m+ 3M2xm− 2M2axm

−M3x2 + [p2T+ L2

a(m2)] (m+M) +M3x

)

+M2am(κa − 1)−MM2

a(1 + κa)x

]

,

(C5)

I ′ dip1 =

∫

dl′T

(2π)2(l′

T+ pT ) · pT

p2T

1

(l′T+ pT )2 [l′ 2T + L2

a(Λ2a)]

3

=

∫ ∞

0

d|l′T||l′

T|

(2π)2p2T

1

[l′ 2T

+ L2a(Λ

2a)]

3

∫ 2π

0

dθ|l′

T||pT | cos θ + p2

T

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=

∫ |pT |

0

d|l′T||l′

T|

(2π)p2T

1

[l′ 2T

+ L2a(Λ

2a)]

3=

1

4πp2T

∫ L2a(Λ

2a)+p2

T

L2a(Λ

2a)

dy

y3

=1

8πp2T

[

1

L4a(Λ

2a)

− 1

[L2a(Λ

2a) + p2

T]2

]

=2L2

a(Λ2a) + p2

T

8πL4a(Λ

2a) [L

2a(Λ

2a) + p2

T]2,

I ′ dip2 =

∫

dl′T

(2π)2(l′

T+ pT )

2

(l′T+ pT )2 [l′ 2T + L2

a(Λ2a)]

3

=

∫ ∞

0

d|l′T||l′

T|

(2π)21

[l′ 2T

+ L2a(Λ

2a)]

3

∫ 2π

0

dθ =1

4π

∫ ∞

L2a(Λ

2a)

dy

y3

=1

8πL4a(Λ

2a), (C6)

I ′ dip3 =

∫

d2l′T

(2π)2[(l′

T+ pT ) · pT ]

2

p2T

1

(l′T+ pT )2 [l′ 2T + L2

a(Λ2a)]

3

=

∫ ∞

0

d|l′T||l′

T|

(2π)2p2T

1

[l′ 2T

+ L2a(Λ

2a)]

3

∫ 2π

0

dθ

[


T

]2

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=1

4πp2T

{

∫ |pT |

0

d|l′T||l′

T|

[l′ 2T

+ L2a(Λ

2a)]

3(2p2

T− l′ 2

T) + p2

T

∫ ∞

|pT |

d|l′T||l′

T|

[l′ 2T

+ L2a(Λ

2a)]

3

}

39

=1

8πp2T

{

∫ L2a(Λ

2a)+p2

T

L2a(Λ

2a)

dy2p2

T+ L2

a(Λ2a)− y

y3+ p2

T

∫ ∞

L2a(Λ

2a)+p2

T

dy

y3

}

=1

8πp2T

{

(

2p 6T+ 3L2

a(Λ2a)p

4T

2L2a(Λ

2a) [L

2a(Λ

2a) + p2

T]2

)

+ p2T

(

1

2 [L2a(Λ

2a) + p2

T]2

)

}

=L2a(Λ

2a) + 2p2

T

16πL4a(Λ

2a) [L

2a(Λ

2a) + p2

T],

I ′ dip4 =

∫

dl′T

(2π)2(l′

T+ pT ) · pT

p2T

(l′T+ pT )

2

(l′T+ pT )2 [l′ 2T + L2

a(Λ2a)]

3

=

∫ ∞

0

d|l′T||l′

T|

(2π)2p2T

1

[l′ 2T

+ L2a(Λ

2a)]

3

∫ 2π

0

dθ(


T

)

=

∫ ∞

0

d|l′T||l′

T|

(2π)

1

[l′ 2T

+ L2a(Λ

2a)]

3=

1

4π

∫ ∞

L2a(Λ

2a)

dy

y3

=1

8πL4a(Λ

2a)

≡ I ′ dip2 ,

I ′ dip5 =

∫

dl′T

(2π)2(l′

T+ pT )

4

(l′T+ pT )2 [l′ 2T + L2

a(Λ2a)]

3

=

∫ ∞

0

d|l′T||l′

T|

(2π)21

[l′ 2T

+ L2a(Λ

2a)]

3

∫ 2π

0

dθ(

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

)

=

∫ ∞

0

d|l′T||l′

T|

(2π)

(l′ 2T

+ p2T)

[l′ 2T

+ L2a(Λ

2a)]

3=

1

4π

∫ ∞

L2a(Λ

2a)

dyy − L2

a(Λ2a) + p2

T

y3

=L2a(Λ

2a) + p2

T

8πL4a(Λ

2a)

,

I ′ dip6 =

∫

dl′T

(2π)2(l′

T+ pT ) · ST

S2T

1

(l′T+ pT )2 [l′ 2T + L2

a(Λ2a)]

3

=

∫ ∞

0

d|l′T||l′

T|

(2π)2p2T

1

[l′ 2T

+ L2a(Λ

2a)]

3

∫ 2π

0

dθ|l′

T||ST | cos[θ + (φ− φS)] + |pT ||ST | cos(φ− φS)

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=1

4πS2T

|ST ||pT |

cos(φ − φS)

∫ L2a(Λ

2a)+p

2T

L2a(Λ

2a)

dy

y3=

1

8πS2T

|ST ||pT |

[

1

L4a(Λ

2a)

− 1

[L2a(Λ

2a) + p2

T]2

]

cos(φ− φS)

=1

8πS2T

(pT · ST )2L2

a(Λ2a) + p2

T

L4a(Λ

2a) [L

2a(Λ

2a) + p2

T]2,

I ′ dip7 =

∫

dl′T

(2π)2[(l′

T+ pT ) · ST ]

2

S2T

1

(l′T+ pT )2 [l′ 2T + L2

a(Λ2a)]

3

=

∫ ∞

0

d|l′T||l′

T|

(2π)2S2T

1

[l′ 2T

+ L2a(Λ

2a)]

3

∫ 2π

0

dθ

[

|l′T||ST | cos[θ + (φ− φS)] + |pT ||ST | cos(φ− φS)

]2

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=1

(2π)2S2

T

{

∫ L2a(Λ

2a)+p2

T

L2a(Λ

2a)

dy

2 y3πS2

T

p2T

[

p2T+(

p2T+ L2

a(Λ2a)− y

)

cos 2(φ− φS)]

+ πS2T

∫ ∞

L2a(Λ

2a)+p2

T

dy

2 y3

}

=1

(2π)2S2T

{[

πS2

Tp2

T

4L4a(Λ

2a) [L

2a(Λ

2a) + p2

T]2

(

(cos 2(φ− φS) + 1)p2T+ (cos 2(φ− φS) + 2)L2

a(Λ2a))

]

+

[

πS2

T

4 [L2a(Λ

2a) + p2

T]2

]}

40

=1

16πL4a(Λ

2a) [L

2a(Λ

2a) + p2

T]

[

L2a(Λ

2a) + p2

T

(

1 + cos 2(φ− φS))]

=1

16πL4a(Λ

2a) [L

2a(Λ

2a) + p2

T]

[

L2a(Λ

2a) + 2

(pT · ST )2

S2T

]

,

(C7)

Iexp2 =

∫

dl′T

(2π)2e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

(l′T+ pT )2 [l′ 2T + L2

a(m2)]

(lT + pT )2

=

∫ ∞

0

d|l′T||l′

T|

(2π)

e−[l′ 2T +L2a(m

2)]/[(1−x)Λ2a]

[l′ 2T

+ L2a(m

2)]

=1

4π

∫ ∞

L2a(m

2)

dye−y/[(1−x)Λ2

a]

y=

1

4πΓ

(

0,L2a(m

2)

(1− x)Λ2a

)

,

Iexp3 =

∫

dl′T

(2π)2e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

(l′T+ pT )2 [l′ 2T + L2

a(m2)]

[(l′T+ pT ) · pT ]

2

p2T

=

∫ ∞

0

d|l′T||l′

T|

(2π)2 p2T

e−[l′ 2T +L2a(m

2)]/[(1−x)Λ2a]

[l′ 2T

+ L2a(m

2)]

∫ 2π

0

dθ

[


T

]2

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=1

4πp2T

{

∫ |pT |

0

d|l′T||l′

T|

[l′ 2T

+ L2a(m

2)](2p2

T− l′ 2

T) e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

+ p2T

∫ ∞

|pT |

d|l′T||l′

T|

[l′ 2T

+ L2a(m

2)]e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

}

=1

8πp2T

{

∫ L2a(m

2)+p2T

L2a(m

2)

dy2p2

T+ L2

a(m2)− y

ye−y/[(1−x)Λ2

a] + p2T

∫ ∞

L2a(m

2)+p2T

dye−y/[(1−x)Λ2

a]

y

}

=1

8πp2T

{

[L2a(m

2) + 2p2T] Γ

(

0,L2a(m

2)

(1− x)Λ2a

)

− [L2a(m

2) + p2T] Γ

(

0,L2a(m

2) + p2T

(1− x)Λ2a

)

+ (1− x) Λ2a

(

e−[L2a(m

2)+p2T ]/[(1−x)Λ2

a] − e−L2a(m

2)/[(1−x)Λ2a])}

,

Iexp4 =

∫

dl′T

(2π)2e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

(l′T+ pT )2 [l′ 2T + L2

a(m2)]

[(l′T+ pT ) · pT ]

2

p2T

(l′T+ pT )

2

=

∫ ∞

0

d|l′T||l′

T|

(2π)2 p2T

e−[l′ 2T +L2a(m

2)]/[(1−x)Λ2a]

[l′ 2T

+ L2a(m

2)]

∫ 2π

0

dθ(


T

)

=

∫ ∞

0

d|l′T||l′

T|

2π

e−[l′ 2T +L2a(m

2)]/[(1−x)Λ2a]

[l′ 2T

+ L2a(m

2)]

=1

4π

∫ ∞

L2a(m

2)

dye−y/[(1−x)Λ2

a]

y=

1

4πΓ

(

0,L2a(m

2)

(1 − x)Λ2a

)

≡ Iexp2 ,

Iexp5 =

∫

dl′T

(2π)2e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

(l′T+ pT )2 [l′ 2T + L2

a(m2)]

(l′T+ pT )

4

=

∫ ∞

0

d|l′T||l′

T|

(2π)2e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

[l′ 2T

+ L2a(m

2)]

∫ 2π

0

dθ(

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

)

=

∫ ∞

0

d|l′T||l′

T|

2π

e−[l′ 2T +L2a(m

2)]/[(1−x)Λ2a]

[l′ 2T

+ L2a(m

2)](l′ 2

T+ p2

T)

=1

4π

∫ ∞

L2a(m

2)

dyy − L2

a(m2) + p2

T

y3e−y/[(1−x)Λ2

a]

41

=1

4π

{

[p2T− L2

a(m2)] Γ

(

0,L2a(m

2)

(1− x)Λ2a

)

+ (1− x) Λ2a e

−L2a(m

2)/[(1−x)Λ2a]

}

,

Iexp6 =

∫

dl′T

(2π)2e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

(l′T+ pT )2 [l′ 2T + L2

a(m2)]

(l′T+ pT ) · ST

S2T

=

∫ ∞

0

d|l′T||l′

T|

(2π)2S2T

e−[l′ 2T +L2a(m

2)]/[(1−x)Λ2a]

[l′ 2T

+ L2a(m

2)]

∫ 2π

0

dθ|l′

T||ST | cos(θ + φ− φS) + |pT ||ST | cos(φ− φS)

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=|pT ||ST | cos(φ− φS)

4πS2Tp2

T

∫ L2a(m

2)+p2T

L2a(m

2)

dye−y/[(1−x)Λ2

a]

y

=pT · ST

4πS2Tp2

T

{

Γ

(

0,L2a(m

2)

(1− x)Λ2a

)

− Γ

(

0,L2a(m

2) + p2T

(1− x)Λ2a

)}

,

Iexp7 =

∫

dl′T

(2π)2e−[l′ 2T +L2

a(m2)]/[(1−x)Λ2

a]

(l′T+ pT )2 [l′ 2T + L2

a(m2)]

[(l′T+ pT ) · ST ]

2

S2T

=

∫ ∞

0

d|l′T||l′

T|

(2π)2S2T

e−[l′ 2T +L2a(m

2)]/[(1−x)Λ2a]

[l′ 2T

+ L2a(m

2)]

∫ 2π

0

dθ|l′

T||ST | cos(θ + φ− φS) + |pT ||ST | cos(φ− φS)

l′ 2T

+ p2T+ 2|l′

T||pT | cos θ

=1

(2π)2S2T

{

∫ L2a(m

2)+p2T

L2a(m

2)

dye−y/[(1−x)Λ2

a]

yπS2

T

p2T

[

p2T+ (p2

T+ L2

a(m2)− y) cos 2(φ− φS)

]

+ πS2T

∫ ∞

L2a(m

2)+p2T

dye−y/[(1−x)Λ2

a]

y

}

=1

8πp2T

{

(1 − x) Λ2a

(

e−[L2a(m

2)+p2T ]/[(1−x)Λ2

a] − e−L2a(m

2)/[(1−x)Λ2a])

cos 2(φ− φS)

+ Γ

(

0,L2a(m

2)

(1− x) Λ2a

)

[

[p2T+ L2

a(m2)] cos 2(φ− φS) + p2

T

]

− Γ

(

0,L2a(m

2) + p2T

(1− x)Λ2a

)

[p2T+ L2

a(m2)] cos 2(φ− φS)

}

. (C8)

[1] J. C. Collins and D. E. Soper, Nucl. Phys. B194, 445 (1982).[2] M. Burkardt, Phys. Rev. D62, 071503 (2000), hep-ph/0005108.[3] M. Diehl, Eur. Phys. J. C25, 223 (2002), hep-ph/0205208.[4] X. Ji, Phys. Rev. Lett. 91, 062001 (2003), hep-ph/0304037.[5] A. V. Belitsky, X. Ji, and F. Yuan, Phys. Rev. D69, 074014 (2004), hep-ph/0307383.[6] S. Boffi and B. Pasquini, Riv. Nuovo Cim. 30, 387 (2007), 0711.2625.[7] X. Ji, J.-P. Ma, and F. Yuan, Phys. Rev. D71, 034005 (2005), hep-ph/0404183.[8] X. Ji, J.-P. Ma, and F. Yuan, Phys. Lett. B597, 299 (2004), hep-ph/0405085.[9] A. Bacchetta, M. Boglione, A. Henneman, and P. J. Mulders, Phys. Rev. Lett. 85, 712 (2000), hep-ph/9912490.

[10] P. V. Pobylitsa, (2003), hep-ph/0301236.[11] S. J. Brodsky and F. Yuan, Phys. Rev. D74, 094018 (2006), hep-ph/0610236.[12] A. Bacchetta, D. Boer, M. Diehl, and P. J. Mulders, (2008), 0803.0227.[13] S. J. Brodsky, D. S. Hwang, and I. Schmidt, Phys. Lett. B530, 99 (2002), hep-ph/0201296.[14] D. Boer and P. J. Mulders, Phys. Rev. D57, 5780 (1998), hep-ph/9711485.[15] A. Bacchetta et al., JHEP 02, 093 (2007), hep-ph/0611265.[16] J. C. Collins, Phys. Lett. B536, 43 (2002), hep-ph/0204004.[17] U. D’Alesio and F. Murgia, (2007), arXiv:0712.4328 [hep-ph].[18] U. D’Alesio and F. Murgia, Phys. Rev. D70, 074009 (2004), hep-ph/0408092.[19] D. W. Sivers, Phys. Rev. D41, 83 (1990).[20] M. Anselmino et al., (2005), hep-ph/0511017.[21] D. Boer and P. J. Mulders, Phys. Rev. D57, 5780 (1998), hep-ph/9711485.

42

[22] B. Zhang, Z. Lu, B.-Q. Ma, and I. Schmidt, Phys. Rev. D77, 054011 (2008), arXiv:0803.1692 [hep-ph].[23] S. J. Brodsky, D. S. Hwang, and I. Schmidt, Nucl. Phys. B642, 344 (2002), hep-ph/0206259.[24] M. Burkardt and D. S. Hwang, Phys. Rev. D69, 074032 (2004), hep-ph/0309072.[25] K. Goeke, S. Meissner, A. Metz, and M. Schlegel, Phys. Lett. B637, 241 (2006), hep-ph/0601133.[26] Z. Lu and I. Schmidt, Phys. Rev. D75, 073008 (2007), hep-ph/0611158.[27] S. Meissner, A. Metz, and K. Goeke, Phys. Rev. D76, 034002 (2007), hep-ph/0703176.[28] J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. D76, 074029 (2007), arXiv:0706.1196 [hep-ph].[29] M. Burkardt and B. Hannafious, Phys. Lett. B658, 130 (2008), arXiv:0705.1573 [hep-ph].[30] L. P. Gamberg, G. R. Goldstein, and K. A. Oganessyan, Phys. Rev. D67, 071504 (2003), hep-ph/0301018.[31] L. P. Gamberg, G. R. Goldstein, and K. A. Oganessyan, Phys. Rev. D68, 051501(R) (2003), hep-ph/0307139.[32] Z. Lu and B.-Q. Ma, Nucl. Phys. A741, 200 (2004), hep-ph/0406171.[33] L. P. Gamberg and G. R. Goldstein, Phys. Lett. B650, 362 (2007), hep-ph/0506127.[34] V. Barone, Z. Lu, and B.-Q. Ma, Eur. Phys. J. C49, 967 (2007), hep-ph/0612350.[35] Z. Lu, B.-Q. Ma, and I. Schmidt, Phys. Rev. D75, 094012 (2007), arXiv:0704.2292 [hep-ph].[36] L. P. Gamberg, G. R. Goldstein, and M. Schlegel, Phys. Rev. D77, 094016 (2008), 0708.0324.[37] A. Bianconi and M. Radici, Phys. Rev. D71, 074014 (2005), hep-ph/0412368.[38] A. Bianconi and M. Radici, Phys. Rev. D72, 074013 (2005), hep-ph/0504261.[39] A. Bianconi and M. Radici, Phys. Rev. D73, 034018 (2006), hep-ph/0512091.[40] A. Bianconi and M. Radici, Phys. Rev. D73, 114002 (2006), hep-ph/0602103.[41] M. Radici, F. Conti, A. Bacchetta, and A. Bianconi, (2007), arXiv:0708.0232 [hep-ph].[42] R. Jakob, P. J. Mulders, and J. Rodrigues, Nucl. Phys. A626, 937 (1997), hep-ph/9704335.[43] B. Pasquini, S. Cazzaniga, and S. Boffi, (2008), 0806.2298.[44] D. Boer, S. J. Brodsky, and D. S. Hwang, Phys. Rev. D67, 054003 (2003), hep-ph/0211110.[45] G. R. Goldstein and L. Gamberg, (2002), hep-ph/0209085.[46] A. Bacchetta, A. Schafer, and J.-J. Yang, Phys. Lett. B578, 109 (2004), hep-ph/0309246.[47] F. Yuan, Phys. Lett. B575, 45 (2003), hep-ph/0308157.[48] I. O. Cherednikov, U. D’Alesio, N. I. Kochelev, and F. Murgia, Phys. Lett. B642, 39 (2006), hep-ph/0606238.[49] A. Courtoy, F. Fratini, S. Scopetta, and V. Vento, (2008), 0801.4347.[50] Z. Lu and B.-Q. Ma, Phys. Rev. D70, 094044 (2004), hep-ph/0411043.[51] S. J. Brodsky, D. S. Hwang, B.-Q. Ma, and I. Schmidt, Nucl. Phys. B593, 311 (2001), hep-th/0003082.[52] G. P. Lepage and S. J. Brodsky, Phys. Rev. D22, 2157 (1980).[53] V. Barone and P. G. Ratcliffe, Transverse Spin Physics (World Scientific, River Edge, USA, 2003).[54] H. Avakian, A. V. Efremov, P. Schweitzer, and F. Yuan, (2008), 0805.3355.[55] D. Boer, P. J. Mulders, and F. Pijlman, Nucl. Phys. B667, 201 (2003), hep-ph/0303034.[56] J. C. Collins and D. E. Soper, Nucl. Phys. B193, 381 (1981).[57] J. A. Robinson and T. G. Rizzo, Phys. Rev. D33, 2608 (1986).[58] M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory (Addison-Wesley, Reading, MA, USA, 1995).[59] R. E. Cutkosky, J. Math. Phys. 1, 429 (1960).[60] A. Kotzinian, (2008), 0806.3804.[61] M. Burkardt, Phys. Rev. D66, 114005 (2002), hep-ph/0209179.[62] A. Bacchetta, U. D’Alesio, M. Diehl, and C. A. Miller, Phys. Rev. D70, 117504 (2004), hep-ph/0410050.[63] ZEUS, S. Chekanov et al., Phys. Rev. D67, 012007 (2003), hep-ex/0208023.[64] M. Gluck, E. Reya, M. Stratmann, and W. Vogelsang, Phys. Rev. D63, 094005 (2001), hep-ph/0011215.[65] Asymmetry Analysis, M. Hirai, S. Kumano, and N. Saito, Phys. Rev. D69, 054021 (2004), hep-ph/0312112.[66] I. C. Cloet, W. Bentz, and A. W. Thomas, Phys. Lett. B659, 214 (2008), 0708.3246.[67] M. Burkardt, Phys. Rev. D69, 091501 (2004), hep-ph/0402014.[68] H. Avakian, S. J. Brodsky, A. Deur, and F. Yuan, Phys. Rev. Lett. 99, 082001 (2007), 0705.1553.[69] H. Mkrtchyan et al., (2007), 0709.3020.[70] E. Leader, A. V. Sidorov, and D. B. Stamenov, Int. J. Mod. Phys. A13, 5573 (1998), hep-ph/9708335.[71] M. Hirai, S. Kumano, and M. Miyama, Comput. Phys. Commun. 111, 150 (1998), hep-ph/9712410.[72] M. Anselmino et al., Phys. Rev. D75, 054032 (2007), hep-ph/0701006.[73] HERMES, A. Airapetian et al., Phys. Rev. Lett. 94, 012002 (2005), hep-ex/0408013.[74] COMPASS, E. S. Ageev et al., Nucl. Phys. B765, 31 (2007), hep-ex/0610068.[75] M. Anselmino et al., (2008), 0807.0173.[76] V. Barone, A. Drago, and P. G. Ratcliffe, Phys. Rept. 359, 1 (2002), hep-ph/0104283.[77] B. Pasquini, M. Pincetti, and S. Boffi, Phys. Rev. D72, 094029 (2005), hep-ph/0510376.[78] M. Wakamatsu, Phys. Lett. B653, 398 (2007), 0705.2917.[79] M. Anselmino et al., (2008), 0805.2677.[80] J. C. Collins et al., (2005), hep-ph/0510342.[81] QCDSF, M. Gockeler et al., Phys. Rev. Lett. 98, 222001 (2007), hep-lat/0612032.

1 2Dipartimento di Fisica Nucleare e Teorica, Universita ...2Dipartimento di Fisica Nucleare e...

Documents

Transcript of 1 2Dipartimento di Fisica Nucleare e Teorica, Universita ...2Dipartimento di Fisica Nucleare e...